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Mathematics LibreTexts

1.E: Relations and Functions (Exercises)

1.1: Sets of Real Numbers and the Cartesian Coordinate Plane

\begin{enumerate}

\item Fill in the chart below:

\begin{center}

\begin{tabular}{|c|c|c|} \hline

Set of Real Numbers & Interval Notation & Region on the Real Number Line \\

\hline

& & \\

\shortstack{$\{x\,|\,-1\leq x< 5\}\) \\ \hfill} & & \\ \hline

& & \\

& \shortstack{$[0,3)\) \\ \hfill} & \\ \hline

& & \\

& &

\begin{mfpic}[10]{-3}{3}{-2}{2}

\tlpointsep{4pt}

\axislabels {x}{{$2 \hspace{4pt} \(} -3, {$7$} 3}

\polyline{(-3,0), (3,0)}

\point[3pt]{(3,0)}

\pointfillfalse

\point[3pt]{(-3,0)}

\end{mfpic} \\

\hline

& & \\

\shortstack{$\{x\,|\, -5 < x \leq 0 \}\) \\ \hfill} & & \\ \hline

& & \\

& \shortstack{$(-3,3)\) \\ \hfill} & \\ \hline

& & \\

& &

\begin{mfpic}[10]{-3}{3}{-2}{2}

\tlpointsep{4pt}

\axislabels {x}{{$5 \hspace{4pt} \(} -3, {$7$} 3}

\polyline{(-3,0), (3,0)}

\point[3pt]{(-3,0), (3,0)}

\end{mfpic} \\

\hline

& & \\

\shortstack{$\{x\,| \, x \leq 3 \}\) \\ \hfill} & & \\ \hline

& & \\

& \shortstack{$(\infty, 9)\) \\ \hfill} & \\ \hline

& & \\

& &

\begin{mfpic}[10]{-3}{3}{-2}{2}

\tlpointsep{4pt}

\axislabels {x}{{$4 \hspace{4pt} \(} -3}

\arrow \polyline{(-3,0), (3,0)}

\pointfillfalse

\point[3pt]{(-3,0)}

\end{mfpic} \\

\hline

& & \\

\shortstack{$\{x\,| \, x \geq -3 \}\) \\ \hfill} & & \\ \hline

\end{tabular}

\end{center}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

In Exercises \ref{findunionintfirst} - \ref{findunionintlast}, find the indicated intersection or union and simplify if possible. Express your answers in interval notation.

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item \((-1,5] \cap [0,8)\) \label{findunionintfirst}

\item \((-1,1) \cup [0,6]$

\item \((-\infty,4]\cap (0,\infty)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item \((-\infty,0) \cap [1,5]$

\item \((-\infty, 0) \cup [1,5]$

\item \((-\infty, 5] \cap [5,8)\) \label{findunionintlast}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

In Exercises \ref{writeintervalfirst} - \ref{writeintervallast}, write the set using interval notation.

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item \(\{x\,|\, x \neq 5 \}\) \label{writeintervalfirst}

\item \(\{x\,|\, x \neq -1 \}$

\item \(\{x\,|\, x \neq -3,\, 4 \}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item \(\{x\,|\, x \neq 0, \, 2 \}$

\item \(\{x\,|\, x \neq 2, \, -2 \}$

\item \(\{x\,|\, x \neq 0,\, \pm 4 \}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item \(\{x\,|\, x \leq -1 \, \text{or} \, x \geq 1 \}$

\item \(\{x\,|\, x < 3 \, \text{or} \, x \geq 2 \}$

\item \(\{x\,|\, x \leq -3 \, \text{or} \, x > 0 \}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item \(\{x\,|\, x \leq 5 \, \text{or} \, x = 6 \}$

\item \(\{x\,|\, x > 2 \, \text{or} \, x = \pm 1 \}$

\item \(\{x\,|\, -3 < x < 3 \, \text{or} \, x = 4 \}\) \label{writeintervallast}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item Plot and label the points \(\;A(-3, -7)\), \(\;B(1.3, -2)\), \(\;C(\pi, \sqrt{10})\), \(\;D(0, 8)\), \(\;E(-5.5, 0)\), \(\;F(-8, 4)\), \(\;G(9.2, -7.8)\) and \(H(7, 5)\) in the Cartesian Coordinate Plane given below.

\label{cartexerciseone}

\begin{center}

\begin{mfpic}[15]{-10}{10}{-10}{10}

\axes

\tlabel[cc](10,-0.5){\scriptsize \(x$}

\tlabel[cc](0.5,10){\scriptsize \(y$}

\xmarks{-9,-8,-7,-6,-5,-4,-3,-2,-1,1,2,3,4,5,6,7,8,9}

\ymarks{-9,-8,-7,-6,-5,-4,-3,-2,-1,1,2,3,4,5,6,7,8,9}

\tlpointsep{5pt}

\scriptsize

\axislabels {x}{{$-9 \hspace{7pt}$} -9, {$-8 \hspace{7pt}$} -8, {$-7 \hspace{7pt}$} -7, {$-6 \hspace{7pt}$} -6, {$-5 \hspace{7pt}$} -5, {$-4 \hspace{7pt}$} -4, {$-3 \hspace{7pt}$} -3, {$-2 \hspace{7pt}$} -2, {$-1 \hspace{7pt}$} -1, {$1$} 1, {$2$} 2, {$3$} 3, {$4$} 4, {$5$} 5, {$6$} 6, {$7$} 7, {$8$} 8, {$9$} 9}

\axislabels {y}{{$-9$} -9, {$-8$} -8, {$-7$} -7, {$-6$} -6, {$-5$} -5, {$-4$} -4, {$-3$} -3, {$-2$} -2, {$-1$} -1, {$1$} 1, {$2$} 2, {$3$} 3, {$4$} 4, {$5$} 5, {$6$} 6, {$7$} 7, {$8$} 8, {$9$} 9}

\normalsize

\end{mfpic}

\end{center}

\item For each point given in Exercise \hspace{-.1in} ~\ref{cartexerciseone} above

\begin{itemize}

\item Identify the quadrant or axis in/on which the point lies.

\item Find the point symmetric to the given point about the \(x$-axis.

\item Find the point symmetric to the given point about the \(y$-axis.

\item Find the point symmetric to the given point about the origin.

\end{itemize}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\pagebreak

In Exercises \ref{distmidfirst} - \ref{distmidlast}, find the distance \(d\) between the points and the midpoint \(M\) of the line segment which connects them.

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item \((1,2)\), \((-3,5)\) \label{distmidfirst}

\item \((3, -10)\), \((-1, 2)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item \(\left( \dfrac{1}{2}, 4\right)\), \(\left(\dfrac{3}{2}, -1\right)$

\item \(\left(- \dfrac{2}{3}, \dfrac{3}{2} \right)\), \(\left(\dfrac{7}{3}, 2\right)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item \(\left( \dfrac{24}{5}, \dfrac{6}{5} \right)\), \(\left( -\dfrac{11}{5}, -\dfrac{19}{5} \right)\).

\item \(\left(\sqrt{2}, \sqrt{3}\right)\), \(\left(-\sqrt{8}, -\sqrt{12}\right)\) \vphantom{$\left( \dfrac{6}{5} \right)$}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item \(\left(2 \sqrt{45}, \sqrt{12} \right)\), \(\left(\sqrt{20}, \sqrt{27} \right)\).

\item \((0, 0)\), \((x, y)\) \label{distmidlast}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item Find all of the points of the form \((x, -1)\) which are \(4\) units from the point \((3,2)\).

\item Find all of the points on the \(y$-axis which are \(5\) units from the point \((-5,3)\).

\item Find all of the points on the \(x$-axis which are \(2\) units from the point \((-1,1)\).

\item Find all of the points of the form \((x,-x)\) which are \(1\) unit from the origin.

\item Let's assume for a moment that we are standing at the origin and the positive \(y$-axis points due North while the positive \(x$-axis points due East. Our Sasquatch-o-meter tells us that Sasquatch is 3 miles West and 4 miles South of our current position. What are the coordinates of his position? How far away is he from us? If he runs 7 miles due East what would his new position be?

\item \label{distanceothercases} Verify the Distance Formula \ref{distanceformula} for the cases when:

\begin{enumerate}

\item The points are arranged vertically. (Hint: Use \(P(a, y_{\mbox{\tiny$0$}})\) and \(Q(a, y_{\mbox{\tiny$1$}})\).)

\item The points are arranged horizontally. (Hint: Use \(P(x_{\mbox{\tiny$0$}}, b)\) and \(Q(x_{\mbox{\tiny$1$}}, b)\).)

\item The points are actually the same point. (You shouldn't need a hint for this one.)

\end{enumerate}

\item \label{verifymidpointformula} Verify the Midpoint Formula by showing the distance between \(P(x_{\mbox{\tiny$1$}}, y_{\mbox{\tiny$1$}})\) and \(M\) and the distance between \(M\) and \(Q(x_{\mbox{\tiny$2$}}, y_{\mbox{\tiny$2$}})\) are both half of the distance between \(P\) and \(Q\).

\item Show that the points \(A\), \(\;B\) and \(C\) below are the vertices of a right triangle.

\begin{multicols}{2}

\begin{enumerate}

\item \(A(-3,2)\), \(\;B(-6,4)\), and \(C(1,8)$

\item \(A(-3, 1)\), \(\;B(4, 0)\) and \(C(0, -3)$

\end{enumerate}

\end{multicols}

\item Find a point \(D(x, y)\) such that the points \(A(-3, 1)\), \(\;B(4, 0)\), \(\;C(0, -3)\) and \(D\) are the corners of a square. Justify your answer.

\item Discuss with your classmates how many numbers are in the interval \((0,1)\).

\enlargethispage{.4in}

\item \label{orderedtripleexercise} The world is not flat.\footnote{There are those who disagree with this statement. Look them up on the Internet some time when you're bored.} Thus the Cartesian Plane cannot possibly be the end of the story. Discuss with your classmates how you would extend Cartesian Coordinates to represent the three dimensional world. What would the Distance and Midpoint formulas look like, assuming those concepts make sense at all?

\end{enumerate}

\newpage

\subsection{Answers}

\begin{enumerate}

\item \(~$

\begin{center}

\begin{tabular}{|c|c|c|} \hline

Set of Real Numbers & Interval Notation & Region on the Real Number Line \\

\hline

& & \\

\shortstack{$\{x\,|\,-1\leq x< 5\}\) \\ \hfill} & \shortstack{$[-1,5)\) \\ \hfill} &

\begin{mfpic}[10]{-3}{3}{-2}{2}

\tlpointsep{4pt}

\axislabels {x}{{$-1 \hspace{8pt} \(} -3, {$5$} 3}

\polyline{(-3,0), (3,0)}

\point[3pt]{(-3,0)}

\pointfillfalse

\point[3pt]{(3,0)}

\end{mfpic} \\

\hline

& & \\

\shortstack{$\{x\,|\,0\leq x < 3\}\) \\ \hfill} & \shortstack{$[0,3)\) \\ \hfill} &

\begin{mfpic}[10]{-3}{3}{-2}{2}

\tlpointsep{4pt}

\axislabels {x}{{$0 \hspace{4pt} \(} -3, {$3$} 3}

\polyline{(-3,0), (3,0)}

\point[3pt]{(-3,0)}

\pointfillfalse

\point[3pt]{(3,0)}

\end{mfpic} \\

\hline

& & \\

\shortstack{$\{x\,|\, 2 < x \leq 7 \}\) \\ \hfill} & \shortstack{$(2,7]\) \\ \hfill} &

\begin{mfpic}[10]{-3}{3}{-2}{2}

\tlpointsep{4pt}

\axislabels {x}{{$2 \hspace{4pt} \(} -3, {$7$} 3}

\polyline{(-3,0), (3,0)}

\point[3pt]{(3,0)}

\pointfillfalse

\point[3pt]{(-3,0)}

\end{mfpic} \\

\hline

& & \\

\shortstack{$\{x\,|\, -5 < x \leq 0 \}\) \\ \hfill} & \shortstack{$(-5,0]\) \\ \hfill} &

\begin{mfpic}[10]{-3}{3}{-2}{2}

\tlpointsep{4pt}

\axislabels {x}{{$-5 \hspace{8pt} \(} -3, {$0$} 3}

\polyline{(-3,0), (3,0)}

\point[3pt]{(3,0)}

\pointfillfalse

\point[3pt]{(-3,0)}

\end{mfpic} \\

\hline

& & \\

\shortstack{$\{x\,|\, -3 < x < 3 \}\) \\ \hfill} & \shortstack{$(-3,3)\) \\ \hfill} &

\begin{mfpic}[10]{-3}{3}{-2}{2}

\tlpointsep{4pt}

\axislabels {x}{{$-3 \hspace{8pt} \(} -3, {$3$} 3}

\polyline{(-3,0), (3,0)}

\pointfillfalse

\point[3pt]{(-3,0), (3,0)}

\end{mfpic} \\

\hline

& & \\

\shortstack{$\{x\,|\,5\leq x \leq 7\}\) \\ \hfill}& \shortstack{$[5,7]\) \\ \hfill} &

\begin{mfpic}[10]{-3}{3}{-2}{2}

\tlpointsep{4pt}

\axislabels {x}{{$5 \hspace{4pt} \(} -3, {$7$} 3}

\polyline{(-3,0), (3,0)}

\point[3pt]{(-3,0), (3,0)}

\end{mfpic} \\

\hline

& & \\

\shortstack{$\{x\,| \, x \leq 3 \}\) \\ \hfill} & \shortstack{$(\infty, 3]\) \\ \hfill} &

\begin{mfpic}[10]{-3}{3}{-2}{2}

\tlpointsep{4pt}

\axislabels {x}{{$3$} 3}

\arrow \polyline{(3,0), (-3,0)}

\point[3pt]{(3,0)}

\end{mfpic} \\

\hline

& & \\

\shortstack{$\{x\,| \, x < 9 \}\) \\ \hfill} & \shortstack{$(\infty, 9)\) \\ \hfill} &

\begin{mfpic}[10]{-3}{3}{-2}{2}

\tlpointsep{4pt}

\axislabels {x}{{$9$} 3}

\arrow \polyline{(3,0), (-3,0)}

\pointfillfalse

\point[3pt]{(3,0)}

\end{mfpic} \\

\hline

& & \\

\shortstack{$\{x\,| \, x > 4 \}\) \\ \hfill} & \shortstack{$(4, \infty)\) \\ \hfill} &

\begin{mfpic}[10]{-3}{3}{-2}{2}

\tlpointsep{4pt}

\axislabels {x}{{$4 \hspace{4pt} \(} -3}

\arrow \polyline{(-3,0), (3,0)}

\pointfillfalse

\point[3pt]{(-3,0)}

\end{mfpic} \\

\hline

& & \\

\shortstack{$\{x\,| \, x \geq -3 \}\) \\ \hfill} & \shortstack{$[-3, \infty)\) \\ \hfill} &

\begin{mfpic}[10]{-3}{3}{-2}{2}

\tlpointsep{4pt}

\axislabels {x}{{$-3 \hspace{8pt} \(} -3}

\arrow \polyline{(-3,0), (3,0)}

\point[3pt]{(-3,0)}

\end{mfpic} \\

\hline

\end{tabular}

\end{center}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item \((-1,5] \cap [0,8) = [0,5]$

\item \((-1,1) \cup [0,6] = (-1,6]$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item \((-\infty,4]\cap (0,\infty) = (0,4]$

\item \((-\infty,0) \cap [1,5] = \emptyset$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item \((-\infty, 0) \cup [1,5] = (-\infty,0) \cup [1,5]$

\item \((-\infty, 5] \cap [5,8) = \left\{ 5\right\}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item \((-\infty, 5) \cup (5, \infty)$

\item \((-\infty, -1) \cup (-1, \infty)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item \((-\infty, -3) \cup (-3, 4)\cup (4, \infty)$

\item \((-\infty, 0) \cup (0, 2)\cup (2, \infty)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item \((-\infty, -2) \cup (-2, 2)\cup (2, \infty)$

\item \((-\infty, -4) \cup (-4, 0) \cup (0, 4) \cup (4, \infty)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item \((-\infty, -1] \cup [1, \infty)$

\item \((-\infty, 3) \cup [2, \infty)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item \((-\infty, -3] \cup (0, \infty)$

\item \((-\infty, 5] \cup \{6\}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item \(\{-1\} \cup \{1\} \cup (2, \infty)$

\item \((-3,3) \cup \{4\}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item The required points \(\;A(-3, -7)\), \(\;B(1.3, -2)\), \(\;C(\pi, \sqrt{10})\), \(\;D(0, 8)\), \(\;E(-5.5, 0)\), \(\;F(-8, 4)\), \(\;G(9.2, -7.8)\), and \(H(7, 5)\) are plotted in the Cartesian Coordinate Plane below.

\begin{center}

\begin{mfpic}[20]{-10}{10}{-10}{10}

\axes

\tlabel[cc](10,-0.5){\scriptsize \(x$}

\tlabel[cc](0.5,10){\scriptsize \(y$}

\xmarks{-9,-8,-7,-6,-5,-4,-3,-2,-1,1,2,3,4,5,6,7,8,9}

\ymarks{-9,-8,-7,-6,-5,-4,-3,-2,-1,1,2,3,4,5,6,7,8,9}

\gfill \circle{(-3, -7),0.1}

\tlabel[cc](-3, -7.75){$A(-3,-7)$}

\gfill \circle{(1.3,-2),0.1}

\tlabel[cc](1.5, -2.5){$B(1.3, -2)$}

\gfill \circle{(3.14159, 3.16228),0.1}

\tlabel[cc](3.14, 2.7){$C(\pi, \sqrt{10})$}

\gfill \circle{(0, 8),0.1}

\tlabel[cc](1.25, 8){$D(0, 8)$}

\gfill \circle{(-5.5,0),0.1}

\tlabel[cc](-5.5, 0.5){$E(-5.5,0)$}

\gfill \circle{(-8,4),0.1}

\tlabel[cc](-8, 3.5){$F(-8, 4)$}

\gfill \circle{(9.2,-7.8),0.1}

\tlabel[cc](9.2, -8.3){$G(9.2, -7.8)$}

\gfill \circle{(7 ,5),0.1}

\tlabel[cc](7, 5.5){$H(7, 5)$}

\tlpointsep{5pt}

\scriptsize

\axislabels {x}{{$-9 \hspace{7pt}$} -9, {$-8 \hspace{7pt}$} -8, {$-7 \hspace{7pt}$} -7, {$-6 \hspace{7pt}$} -6, {$-5 \hspace{7pt}$} -5, {$-4 \hspace{7pt}$} -4, {$-3 \hspace{7pt}$} -3, {$-2 \hspace{7pt}$} -2, {$-1 \hspace{7pt}$} -1, {$1$} 1, {$2$} 2, {$3$} 3, {$4$} 4, {$5$} 5, {$6$} 6, {$7$} 7, {$8$} 8, {$9$} 9}

\axislabels {y}{{$-9$} -9, {$-8$} -8, {$-7$} -7, {$-6$} -6, {$-5$} -5, {$-4$} -4, {$-3$} -3, {$-2$} -2, {$-1$} -1, {$1$} 1, {$2$} 2, {$3$} 3, {$4$} 4, {$5$} 5, {$6$} 6, {$7$} 7, {$8$} 8, {$9$} 9}

\normalsize

\end{mfpic}

\end{center}

\pagebreak

\small %In order to fit everything on one page, we made it smaller.

\item \begin{multicols}{2}

\begin{enumerate}

\item The point \(A(-3, -7)\) is

\begin{itemize}

\item in Quadrant III

\item symmetric about \(x$-axis with \((-3, 7)$

\item symmetric about \(y$-axis with \((3, -7)$

\item symmetric about origin with \((3, 7)$

\end{itemize}

\item The point \(B(1.3, -2)\) is

\begin{itemize}

\item in Quadrant IV

\item symmetric about \(x$-axis with \((1.3, 2)$

\item symmetric about \(y$-axis with \((-1.3, -2)$

\item symmetric about origin with \((-1.3, 2)$

\end{itemize}

\setcounter{HWindent}{\value{enumii}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumii}{\value{HWindent}}

\item The point \(C(\pi, \sqrt{10})\) is

\begin{itemize}

\item in Quadrant I

\item symmetric about \(x$-axis with {\small \((\pi, -\sqrt{10})$}

\item symmetric about \(y$-axis with {\small \((-\pi, \sqrt{10})$}

\item symmetric about origin with {\scriptsize \((-\pi, -\sqrt{10})$}

\end{itemize}

\item The point \(D(0, 8)\) is

\begin{itemize}

\item on the positive \(y$-axis

\item symmetric about \(x$-axis with \((0, -8)$

\item symmetric about \(y$-axis with \((0, 8)$

\item symmetric about origin with \((0, -8)$

\end{itemize}

\setcounter{HWindent}{\value{enumii}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumii}{\value{HWindent}}

\item The point \(E(-5.5, 0)\) is

\begin{itemize}

\item on the negative \(x$-axis

\item symmetric about \(x$-axis with \((-5.5, 0)$

\item symmetric about \(y$-axis with \((5.5, 0)$

\item symmetric about origin with \((5.5, 0)$

\end{itemize}

\item The point \(F(-8, 4)\) is

\begin{itemize}

\item in Quadrant II

\item symmetric about \(x$-axis with \((-8, -4)$

\item symmetric about \(y$-axis with \((8, 4)$

\item symmetric about origin with \((8, -4)$

\end{itemize}

\setcounter{HWindent}{\value{enumii}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumii}{\value{HWindent}}

\item The point \(G(9.2, -7.8)\) is

\begin{itemize}

\item in Quadrant IV

\item symmetric about \(x$-axis with \((9.2, 7.8)$

\item symmetric about \(y$-axis with {\scriptsize \((-9.2, -7.8)$}

\item symmetric about origin with \((-9.2, 7.8)$

\end{itemize}

\item The point \(H(7, 5)\) is

\begin{itemize}

\item in Quadrant I

\item symmetric about \(x$-axis with \((7, -5)$

\item symmetric about \(y$-axis with \((-7, 5)$

\item symmetric about origin with \((-7, -5)$

\end{itemize}

\end{enumerate}

\end{multicols}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item \(d = 5\), \(M = \left(-1, \frac{7}{2} \right)$

\item \(d = 4 \sqrt{10}\), \(M = \left(1, -4 \right)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item \(d = \sqrt{26}\), \(M = \left(1, \frac{3}{2} \right)$

\item \(d= \frac{\sqrt{37}}{2}\), \(M = \left(\frac{5}{6}, \frac{7}{4} \right)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item \(d = \sqrt{74}\), \(M = \left(\frac{13}{10}, -\frac{13}{10} \right)\) \vphantom{$\left( \frac{\sqrt{3}}{2} \right)$}

\item \(d= 3\sqrt{5}\), \(M = \left(-\frac{\sqrt{2}}{2}, -\frac{\sqrt{3}}{2} \right)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item \(d = \sqrt{83}\), \(M = \left(4 \sqrt{5}, \frac{5 \sqrt{3}}{2} \right)$

\item \(d = \sqrt{x^2 + y^2}\), \(M = \left( \frac{x}{2}, \frac{y}{2}\right)\) \vphantom{$\left( \frac{\sqrt{3}}{2} \right)$}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item \((3 + \sqrt{7}, -1)\), \((3-\sqrt{7}, -1)$

\item \((0,3)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item \((-1+\sqrt{3},0)\), \((-1-\sqrt{3},0)\) \vphantom{$\left( \frac{\sqrt{3}}{2} \right)$}

\item \(\left(\frac{\sqrt{2}}{2},-\frac{\sqrt{2}}{2} \right)\), \(\left(-\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2}\right)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item \((-3, -4)\), \(5\) miles, \((4, -4)$

\addtocounter{enumi}{2}

\item \begin{enumerate}

\item The distance from \(A\) to \(B\) is \(|AB| = \sqrt{13}\), the distance from \(A\) to \(C\) is \(|AC| = \sqrt{52}\), and the distance from \(B\) to \(C\) is \(|BC| = \sqrt{65}\). Since \(\left(\sqrt{13}\right)^2 + \left( \sqrt{52} \right)^2 = \left( \sqrt{65} \right)^2\), we are guaranteed by the \href{http://en.wikipedia.org/wiki/Pythago...rline{converse of the Pythagorean Theorem}} that the triangle is a right triangle.

\item Show that \(|AC|^{2} + |BC|^{2} = |AB|^{2}$

\end{enumerate}

\end{enumerate}

\normalsize

\closegraphsfile

1.2: Relations

In Exercises \ref{relationfirst} - \ref{relationlast}, graph the given relation.

\begin{enumerate}

\item \{$(-3, 9)$, $\;(-2, 4)$, $\;(-1, 1)$, $\;(0, 0)$, $\;(1, 1)$, $\;(2, 4)$, $\;(3, 9)\}$ \label{relationfirst}

\item \{$(-2, 0)$, $\;(-1, 1)$, $\;(-1, -1)$, $\;(0, 2)$, $\;(0, -2)$, $\;(1, 3)$, $\;(1, -3)\}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\left\{ \left(m, 2m \right) \, | \, n = 0, \pm 1, \pm 2 \right\}$

\item $\left\{ \left(\frac{6}{k}, k \right) \, | \, k = \pm 1, \pm 2, \pm 3, \pm 4, \pm 5, \pm 6 \right\}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\left\{ \left(n, 4 - n^2\right) \, | \, n = 0, \pm 1, \pm 2 \right\}$

\item $\left\{ \left(\sqrt{j}, j \right) \, | \, j = 0, 1, 4, 9 \right\}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\left\{ \left(x, -2 \right) \, | \, x > -4 \right\}$

\item $\left\{ \left(x, 3 \right) \, | \, x \leq 4 \right\}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\left\{ \left(-1, y \right) \, | \, y > 1 \right\}$

\item $\left\{ \left(2, y \right) \, | \, y \leq 5 \right\}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\{ (-2, y) \, | \, -3 < y \leq 4\}$

\item $\left\{ \left(3,y \right) \, | \, -4 \leq y < 3 \right\}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\{ (x, 2) \, | \, -2 \leq x < 3 \}$

\item $\left\{ \left(x,-3 \right) \, | \, -4 < x \leq 4 \right\}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\{ (x, y) \, | \, x > -2 \}$

\item $\left\{ \left(x,y \right) \, | \, x \leq 3 \right\}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\left\{ \left(x,y \right) \, | \, y < 4 \right\}$

\item $\left\{ \left(x,y \right) \, | \, x \leq 3, \, y < 2 \right\}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\left\{ \left(x,y \right) \, | \, x > 0, \, y < 4 \right\}$

\item $\{ (x, y) \, | \, -\sqrt{2} \leq x \leq \frac{2}{3}, \; \pi < y \leq \frac{9}{2} \}$ \label{relationlast}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

In Exercises \ref{relationsetfirst} - \ref{relationsetlast}, describe the given relation using either the roster or set-builder method.

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $~$ \label{relationsetfirst}

\begin{mfpic}[15]{-5}{2}{-2}{5}

\point[4pt]{(-4, -1), (-2, 1), (0, 3), (1, 4)}

\axes

\tlabel[cc](2,-0.5){\scriptsize $x$}

\tlabel[cc](0.5,5){\scriptsize $y$}

\xmarks{-4,-3,-2,-1,1}

\ymarks{-1,1,2,3,4}

\tlpointsep{5pt}

\scriptsize

\axislabels {x}{{$-4 \hspace{7pt}$} -4, {$-3 \hspace{7pt}$} -3, {$-2 \hspace{7pt}$} -2, {$-1 \hspace{7pt}$} -1, {$1$} 1}

\axislabels {y}{{$-1$} -1, {$1$} 1, {$2$} 2, {$3$} 3, {$4$} 4}

\normalsize

\tcaption{Relation $A$}

\end{mfpic}

\vfill

\columnbreak

\item $~$

\begin{mfpic}[15]{-5}{5}{-1}{4}

\arrow \polyline{(-3,3), (5,3)}

\point[3pt]{(-3,3)}

\axes

\tlabel[cc](5,-0.5){\scriptsize $x$}

\tlabel[cc](0.5,4){\scriptsize $y$}

\xmarks{-4,-3,-2,-1,1,2,3,4}

\ymarks{1,2,3}

\tlpointsep{5pt}

\scriptsize

\axislabels {x}{{$-1 \hspace{7pt}$} -1, {$-2 \hspace{7pt}$} -2, {$-3 \hspace{7pt}$} -3, {$-4 \hspace{7pt}$} -4, {$1$} 1, {$2$} 2, {$3$} 3, {$4$} 4}

\axislabels {y}{{$1$} 1, {$2$} 2, {$3$} 3}

\normalsize

\tcaption{Relation $B$}

\end{mfpic}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\pagebreak

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $~$

\begin{mfpic}[15]{-1}{4}{-4}{6}

\arrow \polyline{(2,-3), (2,5)}

\pointfillfalse

\point[3pt]{(2,-3)}

\axes

\tlabel[cc](4,-0.5){\scriptsize $x$}

\tlabel[cc](0.5,6){\scriptsize $y$}

\xmarks{1,2,3}

\ymarks{-3,-2,-1,1,2,3,4,5}

\tlpointsep{5pt}

\scriptsize

\axislabels {x}{{$1$} 1, {$2$} 2, {$3$} 3}

\axislabels {y}{ {$-3$} -3,{$-2$} -2, {$-1$} -1, {$1$} 1, {$2$} 2, {$3$} 3, {$4$} 4, {$5$} 5}

\normalsize

\tcaption{Relation $C$}

\end{mfpic}

\vfill

\columnbreak

\item $~$

\begin{mfpic}[15]{-4}{1}{-5}{4}

\polyline{(-2,-4), (-2,3)}

\point[3pt]{(-2,-4)}

\pointfillfalse

\point[3pt]{(-2,3)}

\axes

\tlabel[cc](1,-0.5){\scriptsize $x$}

\tlabel[cc](0.5,4){\scriptsize $y$}

\xmarks{-3,-2,-1}

\ymarks{-4,-3,-2,-1,1,2,3}

\tlpointsep{5pt}

\scriptsize

\axislabels {x}{{$-3 \hspace{7pt}$} -3, {$-2 \hspace{7pt}$} -2, {$-1 \hspace{7pt}$} -1}

\axislabels {y}{{$-4$} -4,{$-3$} -3, {$-2$} -2, {$-1$} -1, {$1$} 1, {$2$} 2, {$3$} 3}

\normalsize

\tcaption{Relation $D$}

\end{mfpic}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $~$

\begin{mfpic}[15]{-5}{5}{-1}{4}

\polyline{(-4,2), (3,2)}

\point[3pt]{(-4,2)}

\pointfillfalse

\point[3pt]{(3,2)}

\axes

\tlabel[cc](5,-0.5){\scriptsize $x$}

\tlabel[cc](0.5,4){\scriptsize $y$}

\xmarks{-4,-3,-2,-1,1,2,3,4}

\ymarks{1,2,3}

\tlpointsep{5pt}

\scriptsize

\axislabels {x}{{$-4 \hspace{7pt}$} -4,{$-3 \hspace{7pt}$} -3, {$-2 \hspace{7pt}$} -2, {$-1 \hspace{7pt}$} -1, {$1$} 1, {$2$} 2, {$3$} 3, {$4$} 4}

\axislabels {y}{{$1$} 1, {$2$} 2, {$3$} 3}

\normalsize

\tcaption{Relation $E$}

\end{mfpic}

\vfill

\columnbreak

\item $~$

\begin{mfpic}[15]{-4}{4}{-1}{5}

\fillcolor[gray]{.7}

\gfill \rect{(-4,0), (3.75,4.75)}

\axes

\tlabel[cc](4,-0.5){\scriptsize $x$}

\tlabel[cc](0.5,5){\scriptsize $y$}

\xmarks{-3,-2,-1,1,2,3}

\ymarks{1,2,3,4}

\tlpointsep{5pt}

\scriptsize

\axislabels {x}{{$-3 \hspace{7pt}$} -3,{$-2 \hspace{7pt}$} -2, {$-1 \hspace{7pt}$} -1, {$1$} 1, {$2$} 2, {$3$} 3}

\axislabels {y}{ {$1$} 1, {$2$} 2, {$3$} 3, , {$4$} 4}

\normalsize

\tcaption{Relation $F$}

\end{mfpic}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $~$

\begin{mfpic}[15]{-4}{4}{-4}{4}

\fillcolor[gray]{.7}

\gfill \rect{(-1.97,-3.75), (3.75,3.75)}

\arrow \reverse \arrow \dashed \polyline{(-2,-4), (-2,4)}

\axes

\tlabel[cc](4,-0.5){\scriptsize $x$}

\tlabel[cc](0.5,4){\scriptsize $y$}

\xmarks{-3,-2,-1,1,2,3}

\ymarks{-3,-2,-1,1,2,3}

\tlpointsep{5pt}

\scriptsize

\axislabels {x}{{$-3 \hspace{7pt}$} -3,{$-2 \hspace{7pt}$} -2,{$-1 \hspace{7pt}$} -1,{$1$} 1,{$2$} 2,{$3$} 3}

\axislabels {y}{ {$-3$} -3,{$-2$} -2, {$-1$} -1, {$1$} 1, {$2$} 2, {$3$} 3}

\normalsize

\tcaption{Relation $G$}

\end{mfpic}

\vfill

\columnbreak

\item $~$

\begin{mfpic}[15]{-4.5}{4}{-4}{4}

\fillcolor[gray]{.7}

\gfill \rect{(-2.97,-3.75), (1.97,3.75)}

\arrow \reverse \arrow \dashed \polyline{(-3,-4), (-3,4)}

\arrow \reverse \arrow \polyline{(2,-4), (2,4)}

\axes

\tlabel[cc](4,-0.5){\scriptsize $x$}

\tlabel[cc](0.5,4){\scriptsize $y$}

\xmarks{-4,-3,-2,-1,1,2,3}

\ymarks{-3,-2,-1,1,2,3}

\tlpointsep{5pt}

\scriptsize

\axislabels {x}{{$-4 \hspace{7pt}$} -4,{$-3 \hspace{7pt}$} -3,{$-2 \hspace{7pt}$} -2,{$-1 \hspace{7pt}$} -1,{$1$} 1,{$2$} 2,{$3$} 3}

\axislabels {y}{ {$-3$} -3,{$-2$} -2, {$-1$} -1, {$1$} 1, {$2$} 2, {$3$} 3}

\normalsize

\tcaption{Relation $H$}

\end{mfpic}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\pagebreak

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $~$

\begin{mfpic}[15]{-1.5}{6}{-1.5}{6}

\fillcolor[gray]{.7}

\gfill \rect{(0,0), (5.75,5.75)}

\axes

\tlabel[cc](6,-0.5){\scriptsize $x$}

\tlabel[cc](0.5,6){\scriptsize $y$}

\xmarks{-1,1,2,3,4,5}

\ymarks{-1,1,2,3,4,5}

\tlpointsep{5pt}

\scriptsize

\axislabels {x}{ {$-1 \hspace{7pt}$} -1, {$1$} 1, {$2$} 2, {$3$} 3, {$4$} 4, {$5$} 5}

\axislabels {y}{ {$-1$} -1, {$1$} 1, {$2$} 2, {$3$} 3, {$4$} 4, {$5$} 5}

\normalsize

\tcaption{Relation $I$}

\end{mfpic}

\vfill

\columnbreak

\item $~$ \label{relationsetlast}

\begin{mfpic}[15]{-4.5}{5.5}{-4}{3}

\fillcolor[gray]{.7}

\gfill \rect{(-3.97, -2.97), (4.97, 1.97)}

\dashed \polyline{(-4, -3), (-4, 2)}

\dashed \polyline{(-4, 2), (5, 2)}

\dashed \polyline{(5, 2), (5, -3)}

\dashed \polyline{(5, -3), (-4, -3)}

\axes

\tlabel[cc](5.5,-0.5){\scriptsize $x$}

\tlabel[cc](0.5,3){\scriptsize $y$}

\xmarks{-4,-3,-2,-1,1,2,3,4,5}

\ymarks{-3,-2,-1,1,2}

\tlpointsep{5pt}

\scriptsize

\axislabels {x}{{$-4 \hspace{7pt}$} -4, {$-3 \hspace{7pt}$} -3, {$-2 \hspace{7pt}$} -2, {$-1 \hspace{7pt}$} -1, {$1$} 1, {$2$} 2, {$3$} 3, {$4$} 4, {$5$} 5}

\axislabels {y}{{$-3$} -3, {$-2$} -2, {$-1$} -1, {$1$} 1, {$2$} 2}

\normalsize

\tcaption{Relation $J$}

\end{mfpic}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

In Exercises \ref{graphlinefirst} - \ref{graphlinelast}, graph the given line.

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $x = -2$ \label{graphlinefirst}

\item $x = 3$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $y = 3$

\item $y = -2$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $x=0$

\item $y=0$ \label{graphlinelast}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

Some relations are fairly easy to describe in words or with the roster method but are rather difficult, if not impossible, to graph. Discuss with your classmates how you might graph the relations given in Exercises \ref{cannotgraphfirst} - \ref{cannotgraphlast}. Please note that in the notation below we are using the \index{ellipsis (\ldots)} ellipsis, \ldots, to denote that the list does not end, but rather, continues to follow the established pattern indefinitely. For the relations in Exercises \ref{cannotgraphfirst} and \ref{cannotgraphsecond}, give two examples of points which belong to the relation and two points which do not belong to the relation.

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\{(x, y) \, | \, x \mbox{ is an odd integer, and } y \mbox{ is an even integer.}\}$ \label{cannotgraphfirst}

\item $\{(x, 1) \, | \, x \mbox{ is an irrational number }\}$ \label{cannotgraphsecond}

\item $\{(1, 0), (2, 1), (4, 2), (8, 3), (16, 4), (32, 5), \ldots \}$

\item $\{\ldots, (-3, 9), (-2, 4), (-1, 1), (0, 0), (1, 1), (2, 4), (3, 9), \ldots \}$ \label{cannotgraphlast}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

For each equation given in Exercises \ref{oldonethreefirst} - \ref{oldonethreelast}:

\begin{itemize}

\item Find the $x$- and $y$-intercept(s) of the graph, if any exist.

\item Follow the procedure in Example \hspace{-.1in} ~\ref{firstequgraph} to create a table of sample points on the graph of the equation.

\item Plot the sample points and create a rough sketch of the graph of the equation.

\item Test for symmetry. If the equation appears to fail any of the symmetry tests, find a point on the graph of the equation whose reflection fails to be on the graph as was done at the end of Example \ref{secondequgraph}

\end{itemize}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $y = x^{2} + 1$ \label{oldonethreefirst}

\item $y = x^2-2x-8$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $y = x^{3} - x$

\item $y = \frac{x^3}{4} - 3x$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $y = \sqrt{x - 2}$

\item $y = 2 \sqrt{x+4} - 2$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $3x - y = 7$

\item $3x-2y = 10$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $(x+2)^2+y^2 = 16$

\item $x^{2} - y^{2} = 1$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $4y^2 - 9x^2 = 36$

\item $x^{3}y = -4$ \label{oldonethreelast}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

The procedures which we have outlined in the Examples of this section and used in Exercises \ref{oldonethreefirst} - \ref{oldonethreelast} all rely on the fact that the equations were ``well-behaved''. Not everything in Mathematics is quite so tame, as the following equations will show you. Discuss with your classmates how you might approach graphing the equations given in Exercises \ref{listofcurvesfirst} - \ref{listofcurveslast}. What difficulties arise when trying to apply the various tests and procedures given in this section? For more information, including pictures of the curves, each curve name is a link to its page at www.wikipedia.org. For a much longer list of fascinating curves, click \href{http://en.wikipedia.org/wiki/List_of...derline{here}}.

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item \label{listofcurvesfirst} $x^{3} + y^{3} - 3xy = 0\;$ \href{http://en.wikipedia.org/wiki/Folium_...derline{Folium of Descartes}}

\item $x^{4} = x^{2} + y^{2}\;$ \href{http://en.wikipedia.org/wiki/Kampyle...erline{Kampyle of Eudoxus}}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $y^{2} = x^{3} + 3x^{2}\;$ \href{http://en.wikipedia.org/wiki/Tschirn...{Tschirnhausen cubic}}

\item \label{listofcurveslast} $(x^{2} + y^{2})^{2} = x^{3} + y^{3}\;$ \href{http://en.wikipedia.org/wiki/Crooked...erline{Crooked egg}}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item With the help of your classmates, find examples of equations whose graphs possess

\begin{itemize}

\item symmetry about the $x$-axis only

\item symmetry about the $y$-axis only

\item symmetry about the origin only

\item symmetry about the $x$-axis, $y$-axis, and origin

\end{itemize}

Can you find an example of an equation whose graph possesses exactly \textit{two} of the symmetries listed above? Why or why not?

\end{enumerate}

\subsection{Answers}

\begin{multicols}{2}

\begin{enumerate}

\item $~$

\begin{mfpic}[13]{-4}{4}{-1}{10}

\point[4pt]{(-3, 9), (-2, 4), (-1, 1), (0, 0), (1, 1), (2, 4), (3, 9)}

\axes

\tlabel[cc](4,-0.5){\scriptsize $x$}

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\item $~$

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\item $A = \{(-4, -1), (-2, 1), (0, 3), (1, 4)\}$

\item $B = \left\{ \left(x,3 \right) \, | \, x \geq -3 \right\}$

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\item $C = \{ \left(2,y) \, | \, y > -3 \right\}$

\item $D = \{ \left(-2,y) \, | \, -4 \leq y < 3 \right\}$

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\item $E = \left\{ \left(x,2 \right) \, | \, -4 < x \leq 3 \right\}$

\item $F = \{ \left(x,y) \, | \, y \geq 0 \right\}$

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\item $G = \left\{ \left(x,y \right) \, | \, x > -2 \right\}$

\item $H = \left\{ \left(x,y \right) \, | \, -3 < x \leq 2 \right\}$

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\item $I = \{ \left(x,y) \, | \, x \geq 0, \! y \geq 0\right\}$

\item $J = \{(x, y) \, | \, -4 < x < 5, \; -3 < y < 2\}$

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\arrow \reverse \arrow \polyline{(0,4), (0,-4)}

\normalsize

\tcaption{The line $x=0$ is the $y$-axis}

\end{mfpic}

\vfill

\columnbreak

\item $~$

\begin{mfpic}[15]{-4}{4}{-4}{4}

\tlabel[cc](4,-0.5){\scriptsize $x$}

\tlabel[cc](0.5,4){\scriptsize $y$}

\xmarks{-3,-2,-1,1,2,3}

\ymarks{1,2,3,-1,-2,-3}

\tlpointsep{5pt}

\scriptsize

\axislabels {x}{{$-3 \hspace{7pt}$} -3, {$-2 \hspace{7pt}$} -2, {$-1 \hspace{7pt}$} -1, {$1$} 1, {$2$} 2, {$3$} 3}

\axislabels {y}{{$-1$} -1, {$-2$} -2, {$-3$} -3, {$1$} 1, {$2$} 2, {$3$} 3}

\arrow \reverse \arrow \polyline{(0,-4), (0,4)}

\penwd{1.15pt}

\arrow \reverse \arrow \polyline{(4,0), (-4,0)}

\normalsize

\tcaption{The line $y=0$ is the $x$-axis}

\end{mfpic}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\addtocounter{enumi}{4}

\item $y = x^{2} + 1$

\begin{flushleft}

The graph has no $x$-intercepts \smallskip

$y$-intercept: $(0, 1)$ \smallskip

$\begin{array}{|r||c|c|}

\hline

x & y & (x,y) \\ \hline

-2 & 5 & (-2, 5) \\ \hline

-1 & 2 & (-1, 2) \\ \hline

0 & 1 & (0, 1) \\ \hline

1 & 2 & (1, 2) \\ \hline

2 & 5 & (2, 5) \\ \hline

\end{array} $ \smallskip

\begin{mfpic}[10]{-3}{3}{-1}{6}

\point[3pt]{(-2,5), (-1,2), (0,1), (1,2), (2,5)}

\axes

\tlabel[cc](3,-0.5){\scriptsize $x$}

\tlabel[cc](0.5,6){\scriptsize $y$}

\xmarks{-2,-1,1,2}

\ymarks{1,2,3,4,5}

\tlpointsep{4pt}

\axislabels {x}{{\tiny $-2 \hspace{6pt}$} -2, {\tiny $-1 \hspace{6pt}$} -1, {\tiny $1$} 1, {\tiny $2$} 2}

\axislabels {y}{{\tiny $1$} 1, {\tiny $2$} 2, {\tiny $3$} 3, {\tiny $4$} 4, {\tiny $5$} 5}

\arrow \reverse \arrow \function{-2.3, 2.3, 0.1}{x**2+1}

\end{mfpic}

\smallskip

The graph is not symmetric about the $x$-axis (e.g. $(2, 5)$ is on the graph but $(2, -5)$ is not) \smallskip

The graph is symmetric about the $y$-axis \smallskip

The graph is not symmetric about the origin (e.g. $(2, 5)$ is on the graph but $(-2, -5)$ is not)

\end{flushleft}

\vfill

\columnbreak

\item $y = x^{2} - 2x - 8$

\begin{flushleft}

$x$-intercepts: $(4,0)$, $(-2,0)$ \smallskip

$y$-intercept: $(0, -8)$ \smallskip

$\begin{array}{|r||c|c|}

\hline

x & y & (x,y) \\ \hline

-3 & 7 & (-3,7) \\ \hline

-2 & 0 & (-2, 0) \\ \hline

-1 & -5 & (-1, -5) \\ \hline

0 & -8 & (0, -8) \\ \hline

1 & -9 & (1, -9) \\ \hline

2 & -8 & (2, -8) \\ \hline

3 & -5 & (3,-5) \\ \hline

4 & 0 & (4,0) \\ \hline

5 & 7 & (5,7) \\ \hline

\end{array}$ \smallskip

\begin{mfpic}[7]{-4}{6}{-10}{8}

\point[3pt]{(-3,7), (-2,0), (-1,-5), (0,-8), (1,-9), (2,-8), (3,-5), (4,0), (5,7)}

\axes

\tlabel[cc](6,-0.5){\scriptsize $x$}

\tlabel[cc](0.5,8){\scriptsize $y$}

\xmarks{-3,-2,-1,1,2,3,4,5}

\ymarks{-9,-8,-7,-6,-5,-4,-3,-2,-1,1,2,3,4,5,6,7}

\tlpointsep{4pt}

\axislabels {x}{{\tiny $-3 \hspace{6pt}$} -3,{\tiny $-2 \hspace{6pt}$} -2, {\tiny $-1 \hspace{6pt}$} -1, {\tiny $1$} 1, {\tiny $2$} 2, {\tiny $3$} 3, {\tiny $4$} 4, {\tiny $5$} 5}

\axislabels {y}{{\tiny $-9$} -9, {\tiny $-8$} -8, {\tiny $-7$} -7, {\tiny $-6$} -6, {\tiny $-5$} -5, {\tiny $-4$} -4, {\tiny $-3$} -3, {\tiny $-2$} -2, {\tiny $1$} 1, {\tiny $2$} 2, {\tiny $3$} 3, {\tiny $4$} 4, {\tiny $5$} 5, {\tiny $6$} 6, {\tiny $7$} 7}

\arrow \reverse \arrow \function{-3.1, 5.1, 0.1}{x**2-2*x-8}

\end{mfpic}

\smallskip

The graph is not symmetric about the $x$-axis (e.g. $(-3, 7)$ is on the graph but $(-3, -7)$ is not) \smallskip

The graph is not symmetric about the $y$-axis (e.g. $(-3, 7)$ is on the graph but $(3, 7)$ is not) \smallskip

The graph is not symmetric about the origin (e.g. $(-3, 7)$ is on the graph but $(3, -7)$ is not)

\end{flushleft}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\pagebreak

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $y = x^{3} - x$

\begin{flushleft}

$x$-intercepts: $(-1, 0), (0, 0), (1, 0)$ \smallskip

$y$-intercept: $(0, 0)$ \smallskip

$\begin{array}{|r||c|c|}

\hline

x & y & (x,y) \\ \hline

-2 & -6 & (-2, -6) \\ \hline

-1 & 0 & (-1, 0) \\ \hline

0 & 0 & (0, 0) \\ \hline

1 & 0 & (1, 0) \\ \hline

2 & 6 & (2, 6) \\ \hline

\end{array} $ \smallskip

\begin{mfpic}[10]{-3}{3}{-7}{7}

\point[3pt]{(-2,-6), (-1,0), (0,0), (1,0), (2,6)}

\axes

\tlabel[cc](3,-0.5){\scriptsize $x$}

\tlabel[cc](0.5,7){\scriptsize $y$}

\xmarks{-2,-1,1,2}

\ymarks{-6,-5,-4,-3,-2,-1,1,2,3,4,5,6}

\tlpointsep{4pt}

\axislabels {x}{{\tiny $-2 \hspace{6pt}$} -2, {\tiny $-1 \hspace{6pt}$} -1, {\tiny $1$} 1, {\tiny $2$} 2}

\axislabels {y}{{\tiny $-6$} -6,{\tiny $-5$} -5,{\tiny $-4$} -4,{\tiny $-3$} -3,{\tiny $-2$} -2,{\tiny $-1$} -1, {\tiny $1$} 1, {\tiny $2$} 2, {\tiny $3$} 3, {\tiny $4$} 4, {\tiny $5$} 5, {\tiny $6$} 6}

\arrow \reverse \arrow \function{-2.1, 2.1, 0.1}{x**3-x}

\end{mfpic}

\smallskip

The graph is not symmetric about the $x$-axis. (e.g. $(2, 6)$ is on the graph but $(2, -6)$ is not) \smallskip

The graph is not symmetric about the $y$-axis. (e.g. $(2, 6)$ is on the graph but $(-2, 6)$ is not) \smallskip

The graph is symmetric about the origin.

\end{flushleft}

\vfill

\columnbreak

\item $y = \frac{x^3}{4} - 3x$

\begin{flushleft}

$x$-intercepts: $\left(\pm 2\sqrt{3}, 0\right), (0, 0)$ \smallskip

$y$-intercept: $(0,0)$ \smallskip

$\begin{array}{|r||c|c|}

\hline

x & y & (x,y) \\ \hline

-4 & -4 & (-4, -4) \\ \hline

-3 & \frac{9}{4} & \left(-3, \frac{9}{4} \right) \\ \hline

-2 & 4 & (-2, 4) \\ \hline

-1 & \frac{11}{4} & \left(-1, \frac{11}{4}\right) \\ \hline

0 & 0 & (0,0) \\ \hline

1 & -\frac{11}{4} & \left(1, -\frac{11}{4}\right) \\ \hline

2 & -4 & (2, -4) \\ \hline

3 & -\frac{9}{4} & \left(3, -\frac{9}{4} \right) \\ \hline

4 & 4 & (4, 4) \\ \hline

\end{array} $ \smallskip

\begin{mfpic}[10]{-5}{5}{-5}{5}

\point[3pt]{(-4,-4), (-3.4641, 0), (-3, 2.25), (-2,4), (-1,2.75), (0,0), (4,4), (3.4641, 0),(3, -2.25), (2,-4), (1,-2.75)}

\axes

\tlabel[cc](5,-0.5){\scriptsize $x$}

\tlabel[cc](0.5,5){\scriptsize $y$}

\xmarks{-4,-3,-2,-1,1,2,3,4}

\ymarks{-4,-3,-2,-1,1,2,3,4}

\tlpointsep{4pt}

\axislabels {x}{{\tiny $-4 \hspace{6pt}$} -4,{\tiny $-3 \hspace{6pt}$} -3,{\tiny $-2 \hspace{6pt}$} -2,{\tiny $-1 \hspace{6pt}$} -1,{\tiny $1$} 1, {\tiny $2$} 2, {\tiny $3$} 3, {\tiny $4$} 4}

\axislabels {y}{{\tiny $-1$} -1, {\tiny $-2$} -2, {\tiny $-3$} -3, {\tiny $-4$} -4,{\tiny $1$} 1, {\tiny $2$} 2, {\tiny $3$} 3, {\tiny $4$} 4}

\arrow \reverse \arrow \function{-4.1, 4.1, 0.1}{0.25*(x**3)-3*x}

\end{mfpic}

\smallskip

The graph is not symmetric about the $x$-axis (e.g. $(-4, -4)$ is on the graph but $(-4, 4)$ is not) \smallskip

The graph is not symmetric about the $y$-axis (e.g. $(-4, -4)$ is on the graph but $(4, -4)$ is not) \smallskip

The graph is symmetric about the origin

\end{flushleft}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\pagebreak

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $y = \sqrt{x - 2}$

\begin{flushleft}

$x$-intercept: $(2, 0)$ \smallskip

The graph has no $y$-intercepts \smallskip

$\begin{array}{|r||c|c|}

\hline

x & y & (x,y) \\ \hline

2 & 0 & (2, 0) \\ \hline

3 & 1 & (3, 1) \\ \hline

6 & 2 & (6, 2) \\ \hline

11 & 3 & (11, 3) \\ \hline

\end{array} $ \smallskip

\begin{mfpic}[10]{-1}{12}{-1}{4}

\point[3pt]{(2,0), (3,1), (6,2), (11,3)}

\axes

\tlabel[cc](12,-0.5){\scriptsize $x$}

\tlabel[cc](0.5,4){\scriptsize $y$}

\xmarks{1,2,3,4,5,6,7,8,9,10,11}

\ymarks{1,2,3}

\tlpointsep{4pt}

\axislabels {x}{{\tiny $1$} 1, {\tiny $2$} 2, {\tiny $3$} 3, {\tiny $4$} 4, {\tiny $5$} 5, {\tiny $6$} 6, {\tiny $7$} 7, {\tiny $8$} 8, {\tiny $9$} 9, {\tiny $10$} 10, {\tiny $11$} 11}

\axislabels {y}{{\tiny $1$} 1, {\tiny $2$} 2, {\tiny $3$} 3}

\arrow \function{2, 12, 0.1}{sqrt(x - 2)}

\end{mfpic}

\smallskip

The graph is not symmetric about the $x$-axis (e.g. $(3, 1)$ is on the graph but $(3, -1)$ is not) \smallskip

The graph is not symmetric about the $y$-axis (e.g. $(3, 1)$ is on the graph but $(-3, 1)$ is not) \smallskip

The graph is not symmetric about the origin (e.g. $(3, 1)$ is on the graph but $(-3, -1)$ is not)

\end{flushleft}

\vfill

\columnbreak

\item $y = 2 \sqrt{x+4} - 2$

\begin{flushleft}

$x$-intercept: $(-3,0)$ \smallskip

$y$-intercept: $(0,2)$ \smallskip

$\begin{array}{|r||c|c|}

\hline

x & y & (x,y) \\ \hline

-4 & -2 & (-4, -2) \\ \hline

-3 & 0 & (-3,0 ) \\ \hline

-2 & 2 \sqrt{2} -2 & \left(-2, \sqrt{2} -2 \right) \\ \hline

-1 & 2 \sqrt{3} -2 & \left(-2, \sqrt{3} -2 \right) \\ \hline

0 & 2 & (0, 2) \\ \hline

1 & 2 \sqrt{5} -2 & \left(-2, \sqrt{5} -2 \right) \\ \hline

\end{array} $ \smallskip

\begin{mfpic}[10]{-5}{3}{-4}{4}

\point[3pt]{(-4,-2), (-3,0), (-2, 0.8284), (-1, 1.464), (0,2), (1,2.472)}

\axes

\tlabel[cc](3,-0.5){\scriptsize $x$}

\tlabel[cc](0.5,4){\scriptsize $y$}

\xmarks{-4,-3,-2,-1,1,2}

\ymarks{-3,-2,-1,1,2,3}

\tlpointsep{4pt}

\axislabels {x}{{\tiny $-4 \hspace{6pt}$} -4,{\tiny $-3 \hspace{6pt}$} -3, {\tiny $-2 \hspace{6pt}$} -2, {\tiny $-1 \hspace{6pt}$} -1, {\tiny $1$} 1, {\tiny $2$} 2}

\axislabels {y}{{\tiny $-3$} -3, {\tiny $-2$} -2, {\tiny $-1$} -1, {\tiny $1$} 1, {\tiny $2$} 2, {\tiny $3$} 3}

\arrow \function{-4,2,0.1}{2 * sqrt(x+4)-2}

\end{mfpic}

\smallskip

The graph is not symmetric about the $x$-axis (e.g. $(-4, -2)$ is on the graph but $(-4, 2)$ is not) \smallskip

The graph is not symmetric about the $y$-axis (e.g. $(-4, -2)$ is on the graph but $(4, -2)$ is not) \smallskip

The graph is not symmetric about the origin (e.g. $(-4, -2)$ is on the graph but $(4, 2)$ is not)

\end{flushleft}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\pagebreak

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $3x - y = 7$ \\ Re-write as: $y = 3x - 7$.

\begin{flushleft}

$x$-intercept: $(\frac{7}{3}, 0)$ \smallskip

$y$-intercept: $(0, -7)$ \smallskip

$\begin{array}{|r||c|c|}

\hline

x & y & (x,y) \\ \hline

-2 & -13 & (-2,-13) \\ \hline

-1 & -10 & (-1,-10) \\ \hline

0 & -7 & (0, -7) \\ \hline

1 & -4 & (1, -4) \\ \hline

2 & -1 & (2, -1) \\ \hline

3 & 2 & (3, 2) \\ \hline

\end{array} $ \smallskip

\begin{mfpic}[10]{-3}{4}{-14}{4}

\point[3pt]{(-2,-13), (-1,-10), (0, -7), (1, -4), (2, -1), (3, 2)}

\axes

\tlabel[cc](4,-0.5){\scriptsize $x$}

\tlabel[cc](0.5,4){\scriptsize $y$}

\xmarks{-2,-1,1,2,3}

\ymarks{-13,-12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1,1,2,3}

\tlpointsep{4pt}

\axislabels {x}{{\tiny $-2 \hspace{6pt}$} -2, {\tiny $-1 \hspace{6pt}$} -1, {\tiny $1$} 1, {\tiny $2$} 2, {\tiny $3$} 3}

\axislabels {y}{{\tiny $-13$} -13, {\tiny $-12$} -12, {\tiny $-11$} -11, {\tiny $-10$} -10, {\tiny $-9$} -9, {\tiny $-8$} -8, {\tiny $-7$} -7, {\tiny $-6$} -6, {\tiny $-5$} -5, {\tiny $-4$} -4, {\tiny $-3$} -3, {\tiny $-2$} -2, {\tiny $-1$} -1, {\tiny $1$} 1, {\tiny $2$} 2, {\tiny $3$} 3}

\arrow \reverse \arrow \function{-2.2, 3.2, 0.1}{3*x - 7}

\end{mfpic}

\smallskip

The graph is not symmetric about the $x$-axis (e.g. $(3, 2)$ is on the graph but $(3, -2)$ is not) \smallskip

The graph is not symmetric about the $y$-axis (e.g. $(3, 2)$ is on the graph but $(-3, 2)$ is not) \smallskip

The graph is not symmetric about the origin (e.g. $(3, 2)$ is on the graph but $(-3, -2)$ is not)

\end{flushleft}

\vfill

\columnbreak

\item $3x-2y=10$ \\ Re-write as: $y = \frac{3x-10}{2}$.

\begin{flushleft}

$x$-intercepts: $\left(\frac{10}{3}, 0 \right)$ \smallskip

$y$-intercept: $(0, -5)$ \smallskip

$\begin{array}{|r||c|c|}

\hline

x & y & (x,y) \\ \hline

-2 & -8 & (-2, -8) \\ \hline

-1 & -\frac{13}{2} & \left(-1, -\frac{13}{2}\right) \\ \hline

0 & -5 & (0, -5) \\ \hline

1 & -\frac{7}{2} & \left(1, -\frac{7}{2} \right) \\ \hline

2 & -2 & (2, -2) \\ \hline

\end{array} $ \smallskip

\begin{mfpic}[10]{-4}{5}{-10}{3}

\point[3pt]{(-2,-8), (-1,-6.5), (0,-5), (1,-3.5), (2,-2), (3.333,0)}

\axes

\tlabel[cc](5,-0.5){\scriptsize $x$}

\tlabel[cc](0.5,3){\scriptsize $y$}

\xmarks{-3,-2,-1,1,2,3,4}

\ymarks{-9,-8,-7,-6,-5,-4,-3,-2,-1,1,2}

\tlpointsep{4pt}

\axislabels {x}{{\tiny $-3 \hspace{6pt}$} -3,{\tiny $-2 \hspace{6pt}$} -2, {\tiny $-1 \hspace{6pt}$} -1, {\tiny $1$} 1, {\tiny $2$} 2, {\tiny $3$} 3, {\tiny $4$} 4}

\axislabels {y}{{\tiny $-9$} -9,{\tiny $-8$} -8,{\tiny $-7$} -7,{\tiny $-6$} -6,{\tiny $-5$} -5,{\tiny $-4$} -4,{\tiny $-3$} -3,{\tiny $-2$} -2,{\tiny $-1$} -1, {\tiny $1$} 1, {\tiny $2$} 2}

\arrow \reverse \arrow \function{-3, 4.5, 0.1}{(3*x-10)/2}

\end{mfpic}

\smallskip

The graph is not symmetric about the $x$-axis (e.g. $(2, -2)$ is on the graph but $(2,2)$ is not) \smallskip

The graph is not symmetric about the $y$-axis (e.g. $(2, -2)$ is on the graph but $(-2, -2)$ is not) \smallskip

The graph is not symmetric about the origin (e.g. $(2, -2)$ is on the graph but $(-2, 2)$ is not)

\end{flushleft}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\pagebreak

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $(x+2)^2+y^2=16$ \\ Re-write as $y = \pm \sqrt{16-(x+2)^2}$.

\begin{flushleft}

$x$-intercepts: $(-6, 0)$, $(2,0)$ \smallskip

$y$-intercepts: $\left(0, \pm 2\sqrt{3}\right)$ \smallskip

$\begin{array}{|r||c|c|}

\hline

x & y & (x,y) \\ \hline

-6 & 0 & (-6,0) \\ \hline

-4 & \pm 2 \sqrt{3} & \left(-4,\pm 2 \sqrt{3}\right) \\ \hline

-2 & \pm 4 & (-2, \pm 4) \\ \hline

0 & \pm 2 \sqrt{3} & \left(0,\pm 2 \sqrt{3}\right) \\ \hline

2 & 0 & (2, 0) \\ \hline

\end{array} $ \smallskip

\begin{mfpic}[10]{-8}{4}{-6}{6}

\point[3pt]{(-6,0), (-4, 3.4641), (-4, -3.4641), (-2,4), (-2,-4), (0, 3.4641), (0, -3.4641), (2,0) }

\axes

\tlabel[cc](4,-0.5){\scriptsize $x$}

\tlabel[cc](0.5,6){\scriptsize $y$}

\xmarks{-7,-6,-5,-4,-3,-2,-1,1,2,3}

\ymarks{-5,-4,-3,-2,-1,1,2,3,4,5}

\tlpointsep{4pt}

\axislabels {x}{{\tiny $-7 \hspace{6pt}$} -7,{\tiny $-6 \hspace{6pt}$} -6, {\tiny $-5 \hspace{6pt}$} -5,{\tiny $-4 \hspace{6pt}$} -4, {\tiny $-3 \hspace{6pt}$} -3,{\tiny $-2 \hspace{6pt}$} -2, {\tiny $-1 \hspace{6pt}$} -1, {\tiny $1$} 1, {\tiny $2$} 2, {\tiny $3$} 3}

\axislabels {y}{{\tiny $-5$} -5, {\tiny $-4$} -4, {\tiny $-3$} -3, {\tiny $-2$} -2, {\tiny $-1$} -1, {\tiny $1$} 1, {\tiny $2$} 2, {\tiny $3$} 3, {\tiny $4$} 4, {\tiny $5$} 5}

\circle{(-2,0),4}

\end{mfpic}

\smallskip

The graph is symmetric about the $x$-axis \smallskip

The graph is not symmetric about the $y$-axis (e.g. $(-6, 0)$ is on the graph but $(6, 0)$ is not) \smallskip

The graph is not symmetric about the origin (e.g. $(-6, 0)$ is on the graph but $(6, 0)$ is not)

\end{flushleft}

\vfill

\columnbreak

\item $x^{2} - y^{2} = 1$ \\ Re-write as: $y = \pm \sqrt{x^{2} - 1}$.

\begin{flushleft}

$x$-intercepts: $(-1, 0), (1, 0)$ \smallskip

The graph has no $y$-intercepts \smallskip

$\begin{array}{|r||c|c|}

\hline

x & y & (x,y) \\ \hline

-3 & \pm \sqrt{8} & (-3, \pm \sqrt{8}) \\ \hline

-2 & \pm \sqrt{3} & (-2, \pm \sqrt{3}) \\ \hline

-1 & 0 & (-1, 0) \\ \hline

1 & 0 & (1, 0) \\ \hline

2 & \pm \sqrt{3} & (2, \pm \sqrt{3}) \\ \hline

3 & \pm \sqrt{8} & (3, \pm \sqrt{8}) \\ \hline

\end{array} $ \smallskip

\begin{mfpic}[10]{-4}{4}{-4}{4}

\point[3pt]{(-3,2.828), (-3,-2.828),(-2,1.732),(-2,-1.732),(-1,0),(1, 0),(3,2.828),(3,-2.828),(2,1.732),(2, -1.732)}

\axes

\tlabel[cc](4,-0.5){\scriptsize $x$}

\tlabel[cc](0.5,4){\scriptsize $y$}

\xmarks{-3,-2,-1,1,2,3}

\ymarks{-3,-2,-1,1,2,3}

\tlpointsep{4pt}

\axislabels {x}{{\tiny $-3 \hspace{6pt}$} -3, {\tiny $-2 \hspace{6pt}$} -2, {\tiny $-1 \hspace{6pt}$} -1, {\tiny $1$} 1, {\tiny $2$} 2, {\tiny $3$} 3}

\axislabels {y}{{\tiny $-3$} -3, {\tiny $-2$} -2, {\tiny $-1$} -1, {\tiny $1$} 1, {\tiny $2$} 2, {\tiny $3$} 3}

\arrow \reverse \arrow \parafcn{-2,2,0.1}{(cosh(t),sinh(t))}

\arrow \reverse \arrow \parafcn{-2,2,0.1}{(-cosh(t),sinh(t))}

\end{mfpic}

\smallskip

The graph is symmetric about the $x$-axis \smallskip

The graph is symmetric about the $y$-axis \smallskip

The graph is symmetric about the origin

\end{flushleft}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\pagebreak

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $4y^2-9x^2 = 36$ \\

Re-write as: $y = \pm \frac{\sqrt{9x^2+36}}{2}$.

\begin{flushleft}

The graph has no $x$-intercepts \smallskip

$y$-intercepts: $(0, \pm 3)$ \smallskip

$\begin{array}{|r||c|c|}

\hline

x & y & (x,y) \\ \hline

-4 & \pm 3 \sqrt{5} & \left(-4,\pm 3 \sqrt{5}\right) \\ \hline

-2 & \pm 3 \sqrt{2} & \left(-2,\pm 3 \sqrt{2}\right) \\ \hline

0 & \pm 3 & (0, \pm 3) \\ \hline

2 & \pm 3 \sqrt{2} & \left(2,\pm 3 \sqrt{2}\right) \\ \hline

4 & \pm 3 \sqrt{5} & \left(4,\pm 3 \sqrt{5}\right) \\ \hline

\end{array}$ \smallskip

\begin{mfpic}[10]{-5}{5}{-8}{8}

\point[3pt]{(-4, 6.708), (4, 6.708), (-2, 4.243), (2, 4.243), (0,3), (0,-3),(-4, -6.708), (4, -6.708), (-2, -4.243), (2, -4.243) }

\axes

\tlabel[cc](5,-0.5){\scriptsize $x$}

\tlabel[cc](0.5,8){\scriptsize $y$}

\xmarks{-4,-3,-2,-1, 1, 2, 3, 4}

\ymarks{-7,-6,-5,-4,-3,-2,-1,1,2,3,4,5,6,7}

\tlpointsep{4pt}

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\axislabels {y}{{\tiny $-7$} -7, {\tiny $-6$} -6,{\tiny $-5$} -5,{\tiny $-4$} -4,{\tiny $-3$} -3,{\tiny $-2$} -2,{\tiny $-1$} -1,{\tiny $1$} 1,{\tiny $2$} 2,{\tiny $3$} 3,{\tiny $4$} 4,{\tiny $5$} 5,{\tiny $6$} 6,{\tiny $7$} 7 }

\arrow \reverse \arrow \parafcn{-1.6,1.6,0.1}{(2*sinh(t), 3*cosh(t))}

\arrow \reverse \arrow \parafcn{-1.6,1.6,0.1}{(2*sinh(t), 0-3*cosh(t))}

\end{mfpic}

\smallskip

The graph is symmetric about the $x$-axis \smallskip

The graph is symmetric about the $y$-axis \smallskip

The graph is symmetric about the origin

\end{flushleft}

\vfill

\columnbreak

\item $x^{3}y = -4$ \\ Re-write as: $y = -\dfrac{4}{x^{3}}$.

\begin{flushleft}

The graph has no $x$-intercepts \smallskip

The graph has no $y$-intercepts \smallskip

$\begin{array}{|r||c|c|}

\hline

x & y & (x,y) \\ \hline

-2 & \frac{1}{2} & (-2, \frac{1}{2}) \\ \hline

-1 & 4 & (-1, 4) \\ \hline

-\frac{1}{2} & 32 & (-\frac{1}{2}, 32) \\ \hline

\frac{1}{2} & -32 & (\frac{1}{2}, -32)\\ \hline

1 & -4 & (1, -4) \\ \hline

2 & -\frac{1}{2} & (2, -\frac{1}{2}) \\ \hline

\end{array} $ \smallskip

\begin{mfpic}[10]{-5}{5}{-9}{9}

\point[3pt]{(-4,0.125), (-2,1), (-1, 8), (1, -8), (2, -1), (4, -0.125)}

\axes

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\arrow \reverse \arrow \function{0.95, 4.5, 0.1}{-8/(x**3)}

\end{mfpic}

\smallskip

The graph is not symmetric about the $x$-axis (e.g. $(1, -4)$ is on the graph but $(1, 4)$ is not) \smallskip

The graph is not symmetric about the $y$-axis (e.g. $(1, -4)$ is on the graph but $(-1, -4)$ is not)\smallskip

The graph is symmetric about the origin

\end{flushleft}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\closegraphsfile

1.3: Introduction to Functions

In Exercises \ref{setfunctionfirst} - \ref{setfunctionlast}, determine whether or not the relation represents $y$ as a function of $x$. Find the domain and range of those relations which are functions.

\begin{enumerate}

\item \{$(-3, 9)$, $\;(-2, 4)$, $\;(-1, 1)$, $\;(0, 0)$, $\;(1, 1)$, $\;(2, 4)$, $\;(3, 9)\}$ \label{setfunctionfirst}

\item $\left\{ (-3,0), (1,6), (2, -3), (4,2), (-5,6), (4, -9), (6,2) \right\}$

\item $\left\{ (-3,0), (-7,6), (5,5), (6,4), (4,9), (3,0) \right\}$

\item $\left\{ (1,2), (4,4), (9,6), (16,8), (25,10), (36, 12), \ldots \right\}$

\item \{($x, y) \, | \, x$ is an odd integer, and $y$ is an even integer\}

\item \{$(x, 1) \, | \, x$ is an irrational number\}

\item \{$(1, 0)$, $\;(2, 1)$, $\;(4, 2)$, $\;(8, 3)$, $\;(16, 4)$, $\;(32, 5), \;$ \ldots\}

\item \{$\ldots, \; (-3, 9)$, $\;(-2, 4)$, $\;(-1, 1)$, $\;(0, 0)$, $\;(1, 1)$, $\;(2, 4)$, $\;(3, 9), \;$ \ldots\}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\{ (-2, y) \, | \, -3 < y < 4\}$

\item $\{ (x,3) \, | \, -2 \leq x < 4\}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\{ \left(x,x^2\right) \, | \, \text{$x$ is a real number} \}$

\item $\{ \left(x^2,x\right) \, | \, \text{$x$ is a real number} \}$ \label{setfunctionlast}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

In Exercises \ref{graphfunctionfirst} - \ref{graphfunctionlast}, determine whether or not the relation represents $y$ as a function of $x$. Find the domain and range of those relations which are functions.

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $~$ \vspace{-.1in} \label{graphfunctionfirst}

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\vfill

\columnbreak

\item $~$

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\end{mfpic}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\pagebreak

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\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $~$

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\item $~$

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\arrow \reverse \arrow \parafcn{-2,2,0.1}{(cosh(t),sinh(t))}

\arrow \reverse \arrow \parafcn{-2,2,0.1}{(-cosh(t),sinh(t))}

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\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

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\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $~$

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\point[3pt]{(2,0)}

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\vfill

\columnbreak

\item $~$

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\setcounter{HW}{\value{enumi}}

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\begin{enumerate}

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\item $~$

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\dashed \polyline{(-4, -3), (-4, 2)}

\dashed \polyline{(-4, 2), (5, 2)}

\dashed \polyline{(5, 2), (5, -3)}

\dashed \polyline{(5, -3), (-4, -3)}

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\vfill

\columnbreak

\item $~$

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\function{-5,-1,0.1}{-5 - 6*x - x**2}

\function{-1,3,0.1}{x/4 - 7/4}

\point[3pt]{(-5, 0), (-1, 0)}

\gclear \circle{(-3,4), 0.1}

\circle{(-3,4), 0.1}

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\circle{(-1,-2), 0.1}

\gclear \circle{(3,-1), 0.1}

\circle{(3,-1), 0.1}

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\setcounter{HW}{\value{enumi}}

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\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $~$

\begin{mfpic}[8]{-4}{4}{-6}{10}

\point[3pt]{(-2,6), (1,-3) }

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\vfill

\columnbreak

\item $~$

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\plrfcn{0,180,5}{5*sind 3t}

\end{mfpic}

\setcounter{HW}{\value{enumi}}

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\begin{enumerate}

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\item $~$

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\item $~$

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\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $~$

\begin{mfpic}[15]{-3}{3}{-1}{5}

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\item $~$

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\begin{enumerate}

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\item $~$

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\item $~$

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\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $~$

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\item $~$

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\pagebreak

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $~$

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\vfill

\columnbreak

\item $~$ \label{graphfunctionlast}

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\ymarks{-2,-1,1,2}

\tlpointsep{4pt}

\axislabels {x}{{\tiny $-2 \hspace{6pt}$} -2, {\tiny $-1 \hspace{6pt}$} -1, {\tiny $1$} 1, {\tiny $2$} 2}

\axislabels {y}{{\tiny $1$} 1, {\tiny $2$} 2, {\tiny $-2$} -2, {\tiny $-1$} -1}

\arrow \reverse \arrow \polyline{(-3,2), (3,2)}

\end{mfpic}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

In Exercises \ref{equfunctionfirst} - \ref{equfunctionlast}, determine whether or not the equation represents $y$ as a function of $x$.

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $y = x^{3} - x$ \label{equfunctionfirst}

\item $y = \sqrt{x - 2}$

\item $x^{3}y = -4$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $x^{2} - y^{2} = 1$

\item $y = \dfrac{x}{x^{2} - 9}$

\item $x = -6$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $x = y^2 + 4$

\item $y = x^2 + 4$

\item $x^2 + y^2 = 4$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $y = \sqrt{4-x^2}$

\item $x^2 - y^2 = 4$

\item $x^3 + y^3 = 4$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $2x + 3y = 4$

\item $2xy = 4$

\item $x^2 = y^2$ \label{equfunctionlast}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item Explain why the population $P$ of Sasquatch in a given area is a function of time $t$. What would be the range of this function?

\item Explain why the relation between your classmates and their email addresses may not be a function. What about phone numbers and Social Security Numbers?

\setcounter{HW}{\value{enumi}}

\end{enumerate}

The process given in Example \hspace{-.1in} ~\ref{introfunctionlastexample} for determining whether an equation of a relation represents $y$ as a function of $x$ breaks down if we cannot solve the equation for $y$ in terms of $x$. However, that does not prevent us from proving that an equation fails to represent $y$ as a function of $x$. What we really need is two points with the same $x$-coordinate and different $y$-coordinates which both satisfy the equation so that the graph of the relation would fail the Vertical Line Test \hspace{-.1in} ~\ref{VLT}. Discuss with your classmates how you might find such points for the relations given in Exercises \ref{notfuncequfirst} - \ref{notfuncequlast}.

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $x^{3} + y^{3} - 3xy = 0$ \label{notfuncequfirst}

\item $x^{4} = x^{2} + y^{2}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $y^{2} = x^{3} + 3x^{2}$

\item $(x^{2} + y^{2})^{2} = x^{3} + y^{3}$ \label{notfuncequlast}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\newpage

\subsection{Answers}

\begin{multicols}{2}

\begin{enumerate}

\item Function \\ domain = \{$-3$, $-2$, $-1$, $0$, $1$, $2$ ,$3$\} \\ range = \{$0$, $1$, $4$, $9$\}

\vfill

\columnbreak

\item Not a function

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item Function \\ domain = $\left\{ -7, -3, 3, 4, 5, 6 \right\}$ \\ range = $\left\{ 0,4,5,6,9 \right\}$

\vfill

\columnbreak

\item Function \\ domain = $\left\{ 1, 4, 9, 16, 25, 36, \ldots \right\} \\ = \left\{ x \, | \, \text{$x$ is a perfect square} \right\}$ \\ range = $\left\{ 2, 4, 6, 8, 10, 12, \ldots \right\} \\ = \left\{ y \, | \, \text{$y$ is a positive even integer} \right\}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item Not a function

\vfill

\columnbreak

\item Function \\ domain = \{$x$ \, | \, $x$ is irrational\} \\ range = \{$1$\}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item Function \\ domain = \{$x$ \, | \, $x = 2^{n}$ for some whole number $n$\} \\ range = \{$y$ \, | \, $y$ is any whole number\}

\vfill

\columnbreak

\item Function \\ domain = \{$x$ \, | \, $x$ is any integer\} \\ range = \{$y$ \, | \, $y = n^{2}$ for some integer $n$\}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item Not a function

\vfill

\columnbreak

\item Function \\ domain = $[-2, 4)$, range = \{$3$\}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item Function \\ domain = $(-\infty, \infty)$ \\ range = $[0,\infty)$

\vfill

\columnbreak

\item Not a function

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item Function \\ domain = \{$-4$, $-3$, $-2$, $-1$, $0$, $1$\} \\ range = \{$-1$, $0$, $1$, $2$, $3$, $4$\}

\vfill

\columnbreak

\item Not a function

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item Function \\ domain = $(-\infty, \infty)$ \\ range = $[1, \infty)$

\vfill

\columnbreak

\item Not a function

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item Function \\ domain = $[2, \infty)$ \\ range = $[0, \infty)$

\vfill

\columnbreak

\item Function \\ domain = $(-\infty, \infty)$ \\ range = $(0, 4]$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item Not a function

\vfill

\columnbreak

\item Function \\ domain = $[-5,-3) \cup(-3, 3)$ \\ range = $(-2, -1) \cup [0, 4)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item Function \\ domain = $[-2, \infty)$ \\ range = $[-3, \infty)$

\vfill

\columnbreak

\item Not a function

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item Function \\ domain = $[-5,4)$ \\ range = $[-4,4)$

\vfill

\columnbreak

\item Function \\ domain = $[0,3) \cup (3,6]$ \\ range = $(-4,-1] \cup [0,4]$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item Function \\ domain = $(-\infty, \infty)$ \\ range = $(-\infty, 4]$

\vfill

\columnbreak

\item Function \\ domain = $(-\infty, \infty)$ \\ range = $(-\infty, 4]$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item Function \\ domain = $[-2, \infty)$ \\ range = $(-\infty, 3]$

\vfill

\columnbreak

\item Function \\ domain = $(-\infty, \infty)$ \\ range = $(-\infty, \infty)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item Function \\ domain = $(-\infty, 0] \cup (1, \infty)$ \\ range = $(-\infty, 1] \cup \{ 2\}$

\vfill

\columnbreak

\item Function \\ domain = $[-3,3]$ \\ range = $[-2,2]$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item Not a function

\vfill

\columnbreak

\item Function \\ domain = $(-\infty, \infty)$ \\ range = $\{2\}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item Function

\item Function

\item Function

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item Not a function

\item Function

\item Not a function

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item Not a function

\item Function

\item Not a function

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item Function

\item Not a function

\item Function

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item Function

\item Function

\item Not a function

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\closegraphsfile

1.4: Function Notation

\subsection{Exercises}

In Exercises \ref{buildfunctionfirst} - \ref{buildfunctionlast}, find an expression for $f(x)$ and state its domain.

\begin{enumerate}

\item $f$ is a function that takes a real number $x$ and performs the following three steps in the order given: (1) multiply 2; (2) add 3; (3) divide by 4. \label{buildfunctionfirst}

\item $f$ is a function that takes a real number $x$ and performs the following three steps in the order given: (1) add 3; (2) multiply by 2; (3) divide by 4.

\item $f$ is a function that takes a real number $x$ and performs the following three steps in the order given: (1) divide by 4; (2) add 3; (3) multiply by 2.

\item $f$ is a function that takes a real number $x$ and performs the following three steps in the order given: (1) multiply 2; (2) add 3; (3) take the square root.

\item $f$ is a function that takes a real number $x$ and performs the following three steps in the order given: (1) add 3; (2) multiply 2; (3) take the square root.

\item $f$ is a function that takes a real number $x$ and performs the following three steps in the order given: (1) add 3; (2) take the square root; (3) multiply by 2.

\item $f$ is a function that takes a real number $x$ and performs the following three steps in the order given: (1) take the square root; (2) subtract 13; (3) make the quantity the denominator of a fraction with numerator 4.

\item $f$ is a function that takes a real number $x$ and performs the following three steps in the order given: (1) subtract 13; (2) take the square root; (3) make the quantity the denominator of a fraction with numerator 4.

\item $f$ is a function that takes a real number $x$ and performs the following three steps in the order given: (1) take the square root; (2) make the quantity the denominator of a fraction with numerator 4; (3) subtract 13.

\item $f$ is a function that takes a real number $x$ and performs the following three steps in the order given: (1) make the quantity the denominator of a fraction with numerator 4; (2) take the square root; (3) subtract 13. \label{buildfunctionlast}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

In Exercises \ref{funcnotationbasicfirst} - \ref{funcnotationbasiclast}, use the given function $f$ to find and simplify the following:

\begin{multicols}{3}

\begin{itemize}

\item $f(3)$

\item $f(-1)$

\item $f\left(\frac{3}{2} \right)$

\end{itemize}

\end{multicols}

\begin{multicols}{3}

\begin{itemize}

\item $f(4x)$

\item $4f(x)$

\item $f(-x)$

\end{itemize}

\end{multicols}

\begin{multicols}{3}

\begin{itemize}

\item $f(x-4)$

\item $f(x) - 4$

\item $f\left(x^2\right)$

\end{itemize}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $f(x) = 2x+1$ \label{funcnotationbasicfirst}

\item $f(x) = 3 - 4x$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $f(x) = 2 - x^2$

\item $f(x) = x^2 - 3x + 2$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $f(x) = \dfrac{x}{x-1}$

\item $f(x) = \dfrac{2}{x^{3}}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $f(x) = 6$

\item $f(x) = 0$ \label{funcnotationbasiclast}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

In Exercises \ref{secondfuncnotationbasicfirst} - \ref{secondfuncnotationbasiclast}, use the given function $f$ to find and simplify the following:

\begin{multicols}{3}

\begin{itemize}

\item $f(2)$

\item $f(-2)$

\item $f(2a)$

\end{itemize}

\end{multicols}

\begin{multicols}{3}

\begin{itemize}

\item $2 f(a)$

\item $f(a+2)$

\item $f(a) + f(2)$

\end{itemize}

\end{multicols}

\begin{multicols}{3}

\begin{itemize}

\item $f \left( \frac{2}{a} \right)$

\item $\frac{f(a)}{2}$

\item $f(a + h)$

\end{itemize}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $f(x) = 2x-5$ \label{secondfuncnotationbasicfirst}

\item $f(x) = 5-2x$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $f(x) = 2x^2 - 1$

\item $f(x) = 3x^2+3x-2$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $f(x) = \sqrt{2x+1}$

\item $f(x) = 117$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $f(x) = \dfrac{x}{2}$

\item $f(x) = \dfrac{2}{x}$ \label{secondfuncnotationbasiclast}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

In Exercises \ref{findzerofuncfirst} - \ref{findzerofunclast}, use the given function $f$ to find $f(0)$ and solve $f(x) = 0$

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $f(x) = 2x - 1$ \label{findzerofuncfirst}

\item $f(x) = 3 - \frac{2}{5} x$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $f(x) = 2x^2 - 6$

\item $f(x) = x^2 - x - 12$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $f(x) = \sqrt{x+4}$

\item $f(x) = \sqrt{1-2x}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $f(x) = \dfrac{3}{4-x}$

\item $f(x) = \dfrac{3x^2-12x}{4-x^2}$ \label{findzerofunclast}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item Let $f(x) = \left\{ \begin{array}{rr} x + 5, & x \leq -3 \\ \sqrt{9-x^2}, & -3 < x \leq 3 \\ -x+5, & x > 3 \\ \end{array} \right.$ Compute the following function values.

\begin{multicols}{3}

\begin{enumerate}

\item $f(-4)$

\item $f(-3)$

\item $f(3)$

\setcounter{HWindent}{\value{enumii}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumii}{\value{HWindent}}

\item $f(3.001)$

\item $f(-3.001)$

\item $f(2)$

\setcounter{HWindent}{\value{enumii}}

\end{enumerate}

\end{multicols}

\item Let ${\displaystyle f(x) = \left\{ \begin{array}{rcl}

x^{2} & \mbox{ if } & x \leq -1\\

\sqrt{1 - x^{2}} & \mbox{ if } & -1 < x \leq 1\\

x & \mbox{ if } & x > 1 \end{array} \right. }$ Compute the following function values.

\begin{multicols}{3}

\begin{enumerate}

\item $f(4)$

\item $f(-3)$

\item $f(1)$

\setcounter{HWindent}{\value{enumii}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumii}{\value{HWindent}}

\item $f(0)$

\item $f(-1)$

\item $f(-0.999)$

\setcounter{HWindent}{\value{enumii}}

\end{enumerate}

\end{multicols}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

In Exercises \ref{finddomainfirst} - \ref{finddomainlast}, find the (implied) domain of the function.

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $f(x) = x^{4} - 13x^{3} + 56x^{2} - 19$ \label{finddomainfirst}

\item $f(x) = x^2 + 4$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $f(x) = \dfrac{x-2}{x+1}$

\item $f(x) = \dfrac{3x}{x^2+x-2}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $f(x) = \dfrac{2x}{x^2+3}$

\item $f(x) = \dfrac{2x}{x^2-3}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $f(x) = \dfrac{x+4}{x^2 - 36}$

\item $f(x) = \dfrac{x-2}{x-2}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $f(x) = \sqrt{3-x}$

\item $f(x) = \sqrt{2x+5}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $f(x) = 9x\sqrt{x+3}$

\item $f(x) = \dfrac{\sqrt{7-x}}{x^2+1}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $f(x) = \sqrt{6x-2}$

\item $f(x) = \dfrac{6}{\sqrt{6x-2}}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $f(x) = \sqrt[3]{6x-2}$

\item $f(x) = \dfrac{6}{4 - \sqrt{6x-2}}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $f(x) = \dfrac{\sqrt{6x-2}}{x^2-36}$

\item $f(x) = \dfrac{\sqrt[3]{6x-2}}{x^2+36}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $s(t) = \dfrac{t}{t - 8}$

\item $Q(r) = \dfrac{\sqrt{r}}{r - 8}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $b(\theta) = \dfrac{\theta}{\sqrt{\theta - 8}}$

\item $A(x) = \sqrt{x - 7} + \sqrt{9 - x}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\alpha(y) = \sqrt[3]{\dfrac{y}{y - 8}}$

\item $g(v) = \dfrac{1}{4 - \dfrac{1}{v^{2}}}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $T(t) = \dfrac{\sqrt{t} - 8}{5-t}$

\item $u(w) = \dfrac{w - 8}{5 - \sqrt{w}}$ \label{finddomainlast}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item The area $A$ enclosed by a square, in square inches, is a function of the length of one of its sides $x$, when measured in inches. This relation is expressed by the formula $A(x) = x^2$ for $x > 0$. Find $A(3)$ and solve $A(x) = 36$. Interpret your answers to each. Why is $x$ restricted to $x > 0$?

\item The area $A$ enclosed by a circle, in square meters, is a function of its radius $r$, when measured in meters. This relation is expressed by the formula $A(r) = \pi r^2$ for $r > 0$. Find $A(2)$ and solve $A(r) = 16\pi$. Interpret your answers to each. Why is $r$ restricted to $r > 0$?

\item The volume $V$ enclosed by a cube, in cubic centimeters, is a function of the length of one of its sides $x$, when measured in centimeters. This relation is expressed by the formula $V(x) = x^3$ for $x > 0$. Find $V(5)$ and solve $V(x) = 27$. Interpret your answers to each. Why is $x$ restricted to $x > 0$?

\item The volume $V$ enclosed by a sphere, in cubic feet, is a function of the radius of the sphere $r$, when measured in feet. This relation is expressed by the formula $V(r) =\frac{4\pi}{3} r^{3}$ for $r > 0$. Find $V(3)$ and solve $V(r) = \frac{32\pi}{3}$. Interpret your answers to each. Why is $r$ restricted to $r > 0$?

\item The height of an object dropped from the roof of an eight story building is modeled by: $h(t) = -16t^2 + 64$, $0 \leq t \leq 2$. Here, $h$ is the height of the object off the ground, in feet, $t$ seconds after the object is dropped. Find $h(0)$ and solve $h(t) = 0$. Interpret your answers to each. Why is $t$ restricted to $0 \leq t \leq 2$?

\item The temperature $T$ in degrees Fahrenheit $t$ hours after 6 AM is given by $T(t) = -\frac{1}{2} t^2 + 8t+3$ for $0 \leq t \leq 12$. Find and interpret $T(0)$, $T(6)$ and $T(12)$.

\item The function $C(x) = x^2-10x+27$ models the cost, in \textit{hundreds} of dollars, to produce $x$ \textit{thousand} pens. Find and interpret $C(0)$, $C(2)$ and $C(5)$.

\item Using data from the \href{http://www.bts.gov/publications/nati...derline{Bureau of Transportation Statistics}}, the average fuel economy $F$ in miles per gallon for passenger cars in the US can be modeled by $F(t) = -0.0076t^2+0.45t + 16$, $0 \leq t \leq 28$, where $t$ is the number of years since $1980$. Use your calculator to find $F(0)$, $F(14)$ and $F(28)$. Round your answers to two decimal places and interpret your answers to each.

\item The population of Sasquatch in Portage County can be modeled by the function $P(t) = \frac{150t}{t + 15}$, where $t$ represents the number of years since 1803. Find and interpret $P(0)$ and $P(205)$. Discuss with your classmates what the applied domain and range of $P$ should be.

\label{Sasquatchfunc1}

\item \label{piecewiseordering} For $n$ copies of the book \textit{Me and my Sasquatch}, a print on-demand company charges $C(n)$ dollars, where $C(n)$ is determined by the formula \[{\displaystyle C(n) = \left\{ \begin{array}{rcl} 15n & \mbox{ if } & 1 \leq n \leq 25 \\

13.50n & \mbox{ if } & 25 < n \leq 50 \\

12n & \mbox{ if } & n > 50 \\

\end{array} \right. }\]

\begin{enumerate}

\item Find and interpret $C(20)$. % Ans: $C(20) = 300$. It costs $\$300$ for 20 copies of the book.

\item \label{50vs51} How much does it cost to order 50 copies of the book? What about 51 copies? % Ans: $C(50) = 675$, $\$ 675$. $C(51) = 612$, $\$ 612$.

\item Your answer to \ref{50vs51} should get you thinking. Suppose a bookstore estimates it will sell 50 copies of the book. How many books can, in fact, be ordered for the same price as those 50 copies? (Round your answer to a whole number of books.) % Ans: 56 books.

\end{enumerate}

\item \label{piecewiseshipping} An on-line comic book retailer charges shipping costs according to the following formula \[{\displaystyle S(n) = \left\{ \begin{array}{rcl} 1.5 n + 2.5 & \mbox{ if } & 1 \leq n \leq 14 \\

0 & \mbox{ if } & n \geq 15

\end{array} \right. }\]

where $n$ is the number of comic books purchased and $S(n)$ is the shipping cost in dollars.

\begin{enumerate}

\item What is the cost to ship 10 comic books? % Ans: $S(10) = 17.5$, $\$ 17.50$.

\item What is the significance of the formula $S(n) = 0$ for $n \geq 15$? % Ans: There is free shipping on orders of $15$ or more comic books.

\end{enumerate}

\item \label{piecewisemobile} The cost $C$ (in dollars) to talk $m$ minutes a month on a mobile phone plan is modeled by \[{\displaystyle C(m) = \left\{ \begin{array}{rcl} 25 & \mbox{ if } & 0 \leq m \leq 1000 \\

25+0.1(m-1000) & \mbox{ if } & m > 1000

\end{array} \right. }\]

\begin{enumerate}

\item How much does it cost to talk $750$ minutes per month with this plan? % Ans: $C(750) = 25$, $\$ 25$.

\item How much does it cost to talk $20$ hours a month with this plan? % Ans: $C(1200) = 45$, $\$ 45$.

\item Explain the terms of the plan verbally. % Ans: It costs $\$25$ for up to $1000$ minutes and $10$ cents per minute for each minute over $1000$ minutes.

\end{enumerate}

\item \label{greatestinteger} In Section \ref{SetsofNumbers} we defined the set of \index{integer ! greatest integer function}\textbf{integers} as $\mathbb{Z} = \{ \ldots, -3, -2, -1, 0, 1, 2, 3, \ldots\}$.\footnote{The use of the letter $\mathbb{Z}$ for the integers is ostensibly because the German word \textit{zahlen} means `to count.'} The \index{greatest integer function}\textbf{greatest integer of \boldmath{$x$}}, denoted by $\lfloor x \rfloor$, is defined to be the largest integer $k$ with $k \leq x$.

\begin{enumerate}

\item Find $\lfloor 0.785 \rfloor$, $\lfloor 117 \rfloor$, $\lfloor -2.001 \rfloor$, and $\lfloor \pi + 6 \rfloor$

\item Discuss with your classmates how $\lfloor x \rfloor$ may be described as a piecewise defined function.

\smallskip

\textbf{HINT:} There are infinitely many pieces!

\item Is $\lfloor a + b \rfloor = \lfloor a \rfloor + \lfloor b \rfloor$ always true? What if $a$ or $b$ is an integer? Test some values, make a conjecture, and explain your result.

\end{enumerate}

\item We have through our examples tried to convince you that, in general, $f(a + b) \neq f(a) + f(b)$. It has been our experience that students refuse to believe us so we'll try again with a different approach. With the help of your classmates, find a function $f$ for which the following properties are always true.

\begin{enumerate}

\item $f(0) = f(-1 + 1) = f(-1) + f(1)$

\item $f(5) = f(2 + 3) = f(2) + f(3)$

\item $f(-6) = f(0 - 6) = f(0) - f(6)$

\item $f(a + b) = f(a) + f(b)\;$ regardless of what two numbers we give you for $a$ and $b$.

\end{enumerate}

How many functions did you find that failed to satisfy the conditions above? Did $f(x) = x^{2}$ work? What about $f(x) = \sqrt{x}$ or $f(x) = 3x + 7$ or $f(x) = \dfrac{1}{x}$? Did you find an attribute common to those functions that did succeed? You should have, because there is only one extremely special family of functions that actually works here. Thus we return to our previous statement, {\bf in general}, $f(a + b) \neq f(a) + f(b)$.

\end{enumerate}

\subsection{Answers}

\begin{multicols}{2}

\begin{enumerate}

\item $f(x) = \frac{2x+3}{4}$ \\ Domain: $(-\infty, \infty)$

\item $f(x) = \frac{2(x+3)}{4} = \frac{x+3}{2}$ \\ Domain: $(-\infty, \infty)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $f(x) = 2\left(\frac{x}{4} + 3\right) = \frac{1}{2} x + 6$ \\ Domain: $(-\infty, \infty)$

\item $f(x) = \sqrt{2x+3}$ \\ Domain: $\left[ -\frac{3}{2}, \infty \right)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $f(x) = \sqrt{2(x+3)} = \sqrt{2x+6}$ \\ Domain: $[-3, \infty)$

\item $f(x) = 2\sqrt{x+3}$ \\ Domain: $[-3, \infty)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $f(x) = \frac{4}{\sqrt{x} - 13}$ \\ Domain: $[0, 169) \cup (169, \infty)$

\item $f(x) = \frac{4}{\sqrt{x - 13}}$ \\ Domain: $(13, \infty)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $f(x) = \frac{4}{\sqrt{x}} - 13$ \\ Domain: $(0, \infty)$

\item $f(x) = \sqrt{\frac{4}{x}} - 13 = \frac{2}{\sqrt{x}} - 13$ \\ Domain: $(0, \infty)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item For $f(x) = 2x+1$

\begin{multicols}{3}

\begin{itemize}

\item $f(3) = 7$

\item $f(-1) = -1$

\item $f\left(\frac{3}{2} \right) = 4$

\end{itemize}

\end{multicols}

\begin{multicols}{3}

\begin{itemize}

\item $f(4x) = 8x+1$

\item $4f(x) = 8x+4$

\item $f(-x) = -2x+1$

\end{itemize}

\end{multicols}

\begin{multicols}{3}

\begin{itemize}

\item $f(x-4) = 2x-7$

\item $f(x) - 4 = 2x-3$

\item $f\left(x^2\right) = 2x^2+1$

\end{itemize}

\end{multicols}

\item For $f(x) = 3-4x$

\begin{multicols}{3}

\begin{itemize}

\item $f(3) = -9$

\item $f(-1) = 7$

\item $f\left(\frac{3}{2} \right) = -3$

\end{itemize}

\end{multicols}

\begin{multicols}{3}

\begin{itemize}

\item $f(4x) = 3-16x$

\item $4f(x) = 12-16x$

\item $f(-x) = 4x+3$

\end{itemize}

\end{multicols}

\begin{multicols}{3}

\begin{itemize}

\item $f(x-4) = 19-4x$

\item $f(x) - 4 = -4x-1$

\item $f\left(x^2\right) = 3-4x^2$

\end{itemize}

\end{multicols}

\pagebreak

\item For $f(x) = 2 - x^2$

\begin{multicols}{3}

\begin{itemize}

\item $f(3) = -7$

\item $f(-1) = 1$

\item $f\left(\frac{3}{2} \right) = -\frac{1}{4}$

\end{itemize}

\end{multicols}

\begin{multicols}{3}

\begin{itemize}

\item $f(4x) = 2-16x^2$

\item $4f(x) = 8-4x^2$

\item $f(-x) = 2-x^2$

\end{itemize}

\end{multicols}

\begin{multicols}{3}

\begin{itemize}

\item $f(x-4) = -x^2+8x-14$

\item $f(x) - 4 = -x^{2} - 2$

\item $f\left(x^2\right) = 2-x^4$

\end{itemize}

\end{multicols}

\item For $f(x) = x^2 - 3x + 2$

\begin{multicols}{3}

\begin{itemize}

\item $f(3) = 2$

\item $f(-1) = 6$

\item $f\left(\frac{3}{2} \right) = -\frac{1}{4}$

\end{itemize}

\end{multicols}

\begin{multicols}{3}

\begin{itemize}

\item $f(4x) = 16x^2-12x+2$

\item $4f(x) = 4x^2-12x+8$

\item $f(-x) = x^2+3x+2$

\end{itemize}

\end{multicols}

\begin{multicols}{3}

\begin{itemize}

\item $f(x-4) = x^2-11x+30$

\item $f(x) - 4 = x^2-3x-2$

\item $f\left(x^2\right) = x^4-3x^2+2$

\end{itemize}

\end{multicols}

\item For $f(x) = \frac{x}{x-1}$

\begin{multicols}{3}

\begin{itemize}

\item $f(3) = \frac{3}{2}$

\item $f(-1) = \frac{1}{2}$

\item $f\left(\frac{3}{2} \right) = 3$

\end{itemize}

\end{multicols}

\begin{multicols}{3}

\begin{itemize}

\item $f(4x) = \frac{4x}{4x-1}$

\item $4f(x) = \frac{4x}{x-1}$

\item $f(-x) = \frac{x}{x+1}$

\end{itemize}

\end{multicols}

\begin{multicols}{3}

\begin{itemize}

\item $f(x-4) = \frac{x-4}{x-5}$

\vfill

\columnbreak

\item $f(x) - 4 = \frac{x}{x-1} - 4$ \\

$\hphantom{f(x) - 4} = \frac{4-3x}{x-1}$

\vfill

\columnbreak

\item $f\left(x^2\right) = \frac{x^2}{x^2-1}$

\end{itemize}

\end{multicols}

\item For $f(x) = \frac{2}{x^3}$

\begin{multicols}{3}

\begin{itemize}

\item $f(3) = \frac{2}{27}$

\item $f(-1) = -2$

\item $f\left(\frac{3}{2} \right) = \frac{16}{27}$

\end{itemize}

\end{multicols}

\begin{multicols}{3}

\begin{itemize}

\item $f(4x) = \frac{1}{32x^3}$

\item $4f(x) = \frac{8}{x^3}$

\item $f(-x) = -\frac{2}{x^3}$

\end{itemize}

\end{multicols}

\begin{multicols}{3}

\begin{itemize}

\item $f(x-4) = \frac{2}{(x-4)^3}$ \\

$=\frac{2}{x^3-12x^2+48x-64}$

\vfill

\columnbreak

\item $f(x) - 4 = \frac{2}{x^3} - 4$ \\

$\hphantom{f(x) - 4} = \frac{2-4x^3}{x^3}$

\item $f\left(x^2\right) = \frac{2}{x^6}$

\end{itemize}

\end{multicols}

\item For $f(x) = 6$

\begin{multicols}{3}

\begin{itemize}

\item $f(3) = 6$

\item $f(-1) =6$

\item $f\left(\frac{3}{2} \right) = 6$

\end{itemize}

\end{multicols}

\begin{multicols}{3}

\begin{itemize}

\item $f(4x) = 6$

\item $4f(x) = 24$

\item $f(-x) = 6$

\end{itemize}

\end{multicols}

\begin{multicols}{3}

\begin{itemize}

\item $f(x-4) = 6$

\item $f(x) - 4 = 2$

\item $f\left(x^2\right) = 6$

\end{itemize}

\end{multicols}

\pagebreak

\item For $f(x) = 0$

\begin{multicols}{3}

\begin{itemize}

\item $f(3) = 0$

\item $f(-1) =0$

\item $f\left(\frac{3}{2} \right) = 0$

\end{itemize}

\end{multicols}

\begin{multicols}{3}

\begin{itemize}

\item $f(4x) = 0$

\item $4f(x) = 0$

\item $f(-x) = 0$

\end{itemize}

\end{multicols}

\begin{multicols}{3}

\begin{itemize}

\item $f(x-4) = 0$

\item $f(x) - 4 = -4$

\item $f\left(x^2\right) = 0$

\end{itemize}

\end{multicols}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item For $f(x) = 2x-5$

\begin{multicols}{3}

\begin{itemize}

\item $f(2) = -1$

\item $f(-2) = -9$

\item $f(2a) = 4a-5$

\end{itemize}

\end{multicols}

\begin{multicols}{3}

\begin{itemize}

\item $2 f(a) = 4a-10$

\item $f(a+2) = 2a-1$

\item $f(a) + f(2) = 2a-6$

\end{itemize}

\end{multicols}

\begin{multicols}{3}

\begin{itemize}

\item $f \left( \frac{2}{a} \right) = \frac{4}{a} - 5$ \\

$\hphantom{f \left( \frac{2}{a} \right)} = \frac{4-5a}{a}$

\vfill

\columnbreak

\item $\frac{f(a)}{2} =\frac{2a-5}{2}$

\vfill

\columnbreak

\item $f(a + h) = 2a + 2h - 5$

\end{itemize}

\end{multicols}

\item For $f(x) = 5-2x$

\begin{multicols}{3}

\begin{itemize}

\item $f(2) = 1$

\item $f(-2) = 9$

\item $f(2a) = 5-4a$

\end{itemize}

\end{multicols}

\begin{multicols}{3}

\begin{itemize}

\item $2 f(a) = 10-4a$

\item $f(a+2) = 1-2a$

\item $f(a) + f(2) = 6-2a$

\end{itemize}

\end{multicols}

\begin{multicols}{3}

\begin{itemize}

\item $f \left( \frac{2}{a} \right) = 5 - \frac{4}{a}$ \\

$\hphantom{f \left( \frac{2}{a} \right)} = \frac{5a-4}{a}$

\vfill

\columnbreak

\item $\frac{f(a)}{2} = \frac{5-2a}{2}$

\vfill

\columnbreak

\item $f(a + h) = 5-2a-2h$

\end{itemize}

\end{multicols}

\item For $f(x) = 2x^2-1$

\begin{multicols}{3}

\begin{itemize}

\item $f(2) = 7$

\item $f(-2) = 7$

\item $f(2a) = 8a^2-1$

\end{itemize}

\end{multicols}

\begin{multicols}{3}

\begin{itemize}

\item $2 f(a) = 4a^2-2$

\item $f(a+2) = 2a^2+8a+7$

\item $f(a) + f(2) = 2a^2+6$

\end{itemize}

\end{multicols}

\begin{multicols}{3}

\begin{itemize}

\item $f \left( \frac{2}{a} \right) = \frac{8}{a^2} - 1$ \\

$\hphantom{f \left( \frac{2}{a} \right)} = \frac{8-a^2}{a^2}$

\vfill

\columnbreak

\item $\frac{f(a)}{2} = \frac{2a^2-1}{2}$

\vfill

\columnbreak

\item $f(a + h) = 2a^2+4ah+2h^2-1$

\end{itemize}

\end{multicols}

\pagebreak

\item For $f(x) = 3x^2+3x-2$

\begin{multicols}{3}

\begin{itemize}

\item $f(2) = 16$

\item $f(-2) = 4$

\item $f(2a) = 12a^2+6a-2$

\end{itemize}

\end{multicols}

\begin{multicols}{3}

\begin{itemize}

\item $2 f(a) = 6a^2+6a-4$

\item $f(a+2) = 3a^2+15a+16$

\item \small $f(a) + f(2) = 3a^2+3a+14$ \normalsize

\end{itemize}

\end{multicols}

\begin{multicols}{3}

\begin{itemize}

\item $f \left( \frac{2}{a} \right) = \frac{12}{a^2} + \frac{6}{a} - 2$ \\

$\hphantom{f \left( \frac{2}{a} \right)} = \frac{12+6a-2a^2}{a^2}$

\vfill

\columnbreak

\item $\frac{f(a)}{2} = \frac{3a^2+3a-2}{2}$

\vfill

\columnbreak

\item $f(a + h) = 3a^2 + 6ah + 3h^2+3a+3h-2$

\end{itemize}

\end{multicols}

\item For $f(x) = \sqrt{2x+1}$

\begin{multicols}{3}

\begin{itemize}

\item $f(2) = \sqrt{5}$

\item $f(-2)$ is not real

\item $f(2a) = \sqrt{4a+1}$

\end{itemize}

\end{multicols}

\begin{multicols}{3}

\begin{itemize}

\item $2 f(a) = 2\sqrt{2a+1}$

\item $f(a+2) = \sqrt{2a+5}$

\item \small $f(a) + f(2) =\sqrt{2a+1} + \sqrt{5}$ \normalsize

\end{itemize}

\end{multicols}

\begin{multicols}{3}

\begin{itemize}

\item $f \left( \frac{2}{a} \right) = \sqrt{\frac{4}{a} + 1}$ \\

$\hphantom{f \left( \frac{2}{a} \right)} = \sqrt{\frac{a+4}{a}}$

\vfill

\columnbreak

\item $\frac{f(a)}{2} = \frac{\sqrt{2a+1}}{2}$

\vfill

\columnbreak

\item $f(a + h) = \sqrt{2a+2h+1}$

\end{itemize}

\end{multicols}

\item For $f(x) = 117$

\begin{multicols}{3}

\begin{itemize}

\item $f(2) = 117$

\item $f(-2) = 117$

\item $f(2a) = 117$

\end{itemize}

\end{multicols}

\begin{multicols}{3}

\begin{itemize}

\item $2 f(a) = 234$

\item $f(a+2) = 117$

\item $f(a) + f(2) = 234$

\end{itemize}

\end{multicols}

\begin{multicols}{3}

\begin{itemize}

\item $f \left( \frac{2}{a} \right) = 117$

\vfill

\columnbreak

\item $\frac{f(a)}{2} = \frac{117}{2}$

\vfill

\columnbreak

\item $f(a + h) = 117$

\end{itemize}

\end{multicols}

\item For $f(x) = \frac{x}{2}$

\begin{multicols}{3}

\begin{itemize}

\item $f(2) = 1$

\item $f(-2) = -1$

\item $f(2a) = a$

\end{itemize}

\end{multicols}

\begin{multicols}{3}

\begin{itemize}

\item $2 f(a) = a$

\item $f(a+2) = \frac{a+2}{2}$

\vfill

\columnbreak

\item $f(a) + f(2) = \frac{a}{2}+ 1$ \\

$\hphantom{f(a) + f(2)} = \frac{a+2}{2}$

\end{itemize}

\end{multicols}

\begin{multicols}{3}

\begin{itemize}

\item $f \left( \frac{2}{a} \right) = \frac{1}{a}$

\vfill

\columnbreak

\item $\frac{f(a)}{2} = \frac{a}{4}$

\vfill

\columnbreak

\item $f(a + h) = \frac{a+h}{2}$

\end{itemize}

\end{multicols}

\pagebreak

\item For $f(x) = \frac{2}{x}$

\begin{multicols}{3}

\begin{itemize}

\item $f(2) = 1$

\item $f(-2) = -1$

\item $f(2a) = \frac{1}{a}$

\end{itemize}

\end{multicols}

\begin{multicols}{3}

\begin{itemize}

\item $2 f(a) = \frac{4}{a}$

\item $f(a+2) = \frac{2}{a+2}$

\vfill

\columnbreak

\item $f(a) + f(2) = \frac{2}{a}+1$ \\

$\hphantom{f(a)+f(2)}=\frac{a+2}{2}$

\end{itemize}

\end{multicols}

\begin{multicols}{3}

\begin{itemize}

\item $f \left( \frac{2}{a} \right) = a$

\vfill

\columnbreak

\item $\frac{f(a)}{2} = \frac{1}{a}$

\vfill

\columnbreak

\item $f(a + h) = \frac{2}{a+h}$

\end{itemize}

\end{multicols}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item For $f(x) = 2x-1$, $f(0) = -1$ and $f(x) = 0$ when $x = \frac{1}{2}$

\item For $f(x) = 3 - \frac{2}{5} x$, $f(0) = 3$ and $f(x) = 0$ when $x = \frac{15}{2}$

\item For $f(x) = 2x^2-6$, $f(0) = -6$ and $f(x) = 0$ when $x = \pm \sqrt{3}$

\item For $f(x) = x^2-x-12$, $f(0) = -12$ and $f(x) = 0$ when $x = -3$ or $x=4$

\item For $f(x) = \sqrt{x+4}$, $f(0) = 2$ and $f(x) = 0$ when $x =-4$

\item For $f(x) = \sqrt{1-2x}$, $f(0) = 1$ and $f(x) = 0$ when $x = \frac{1}{2}$

\item For $f(x) = \frac{3}{4-x}$, $f(0) = \frac{3}{4}$ and $f(x)$ is never equal to $0$

\item For $f(x) = \frac{3x^2-12x}{4-x^2}$, $f(0) =0$ and $f(x) = 0$ when $x=0$ or $x=4$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item

\begin{multicols}{3}

\begin{enumerate}

\item $f(-4) = 1$

\item $f(-3) = 2$

\item $f(3) = 0$

\setcounter{HWindent}{\value{enumii}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumii}{\value{HWindent}}

\item $f(3.001) = 1.999$

\item $f(-3.001) = 1.999$

\item $f(2) = \sqrt{5}$

\setcounter{HWindent}{\value{enumii}}

\end{enumerate}

\end{multicols}

\item

\begin{multicols}{3}

\begin{enumerate}

\item $f(4) = 4$

\item $f(-3) = 9$

\item $f(1) = 0$

\setcounter{HWindent}{\value{enumii}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumii}{\value{HWindent}}

\item $f(0) = 1$

\item $f(-1) = 1$

\item \small $f(-0.999) \approx 0.0447$ \normalsize

\setcounter{HWindent}{\value{enumii}}

\end{enumerate}

\end{multicols}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $(-\infty, \infty)$

\item $(-\infty, \infty)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $(-\infty, -1) \cup (-1, \infty)$

\item $(-\infty,-2) \cup (-2,1) \cup (1, \infty)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $(-\infty, \infty)$

\item $(-\infty, -\sqrt{3}) \cup (-\sqrt{3}, \sqrt{3}) \cup (\sqrt{3}, \infty)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $(-\infty, -6) \cup (-6,6) \cup (6, \infty)$

\item $(-\infty, 2) \cup (2, \infty)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $(-\infty, 3]$

\item $\left[-\frac{5}{2}, \infty \right)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $[-3, \infty)$

\item $(-\infty, 7]$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\left[ \frac{1}{3}, \infty \right)$

\item $\left( \frac{1}{3}, \infty \right)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $(-\infty, \infty)$

\item $\left[ \frac{1}{3}, 3 \right) \cup (3, \infty)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\left[ \frac{1}{3}, 6 \right) \cup (6, \infty)$

\item $(-\infty, \infty)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $(-\infty, 8) \cup (8, \infty)$

\item $[0, 8) \cup (8, \infty)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $(8, \infty)$

\item $[7, 9]$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $(-\infty, 8) \cup (8, \infty)$

\item $\left( -\infty, -\frac{1}{2} \right) \cup \left( -\frac{1}{2}, 0 \right) \cup \left(0, \frac{1}{2} \right) \cup \left( \frac{1}{2}, \infty\right)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $[0, 5) \cup (5,\infty)$

\item $[0, 25) \cup (25, \infty)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $A(3) = 9$, so the area enclosed by a square with a side of length $3$ inches is $9$ square inches. The solutions to $A(x) = 36$ are $x = \pm 6$. Since $x$ is restricted to $x > 0$, we only keep $x = 6$. This means for the area enclosed by the square to be $36$ square inches, the length of the side needs to be $6$ inches. Since $x$ represents a length, $x > 0$.

\item $A(2) = 4\pi$, so the area enclosed by a circle with radius $2$ meters is $4\pi$ square meters. The solutions to $A(r) = 16\pi$ are $r = \pm 4$. Since $r$ is restricted to $r > 0$, we only keep $r = 4$. This means for the area enclosed by the circle to be $16\pi$ square meters, the radius needs to be $4$ meters. Since $r$ represents a radius (length), $r > 0$.

\item $V(5) = 125$, so the volume enclosed by a cube with a side of length $5$ centimeters is $125$ cubic centimeters. The solution to $V(x) = 27$ is $x = 3$. This means for the volume enclosed by the cube to be $27$ cubic centimeters, the length of the side needs to $3$ centimeters. Since $x$ represents a length, $x > 0$.

\item $V(3) = 36\pi$, so the volume enclosed by a sphere with radius $3$ feet is $36\pi$ cubic feet. The solution to $V(r) = \frac{32\pi}{3}$ is $r = 2$. This means for the volume enclosed by the sphere to be $\frac{32\pi}{3}$ cubic feet, the radius needs to $2$ feet. Since $r$ represents a radius (length), $r > 0$.

\item $h(0) = 64$, so at the moment the object is dropped off the building, the object is $64$ feet off of the ground. The solutions to $h(t) = 0$ are $t = \pm 2$. Since we restrict $0 \leq t \leq 2$, we only keep $t = 2$. This means $2$ seconds after the object is dropped off the building, it is $0$ feet off the ground. Said differently, the object hits the ground after $2$ seconds. The restriction $0 \leq t \leq 2$ restricts the time to be between the moment the object is released and the moment it hits the ground.

\item $T(0) = 3$, so at 6 AM ($0$ hours after 6 AM), it is $3^{\circ}$ Fahrenheit. $T(6) = 33$, so at noon ($6$ hours after 6 AM), the temperature is $33^{\circ}$ Fahrenheit. $T(12) = 27$, so at 6 PM ($12$ hours after 6 AM), it is $27^{\circ}$ Fahrenheit.

\item $C(0) = 27$, so to make $0$ pens, it costs\footnote{This is called the `fixed' or `start-up' cost. We'll revisit this concept on page \pageref{pricerevenuecostprofit}.} $\$ 2700$. $C(2) = 11$, so to make $2000$ pens, it costs $\$1100$. $C(5) = 2$, so to make $5000$ pens, it costs $\$2000$.

\item $F(0) = 16.00$, so in 1980 ($0$ years after 1980), the average fuel economy of passenger cars in the US was $16.00$ miles per gallon. $F(14) = 20.81$, so in 1994 ($14$ years after 1980), the average fuel economy of passenger cars in the US was $20.81$ miles per gallon. $F(28) = 22.64$, so in 2008 ($28$ years after 1980), the average fuel economy of passenger cars in the US was $22.64$ miles per gallon.

\item $P(0) = 0$ which means in 1803 ($0$ years after 1803), there are no Sasquatch in Portage County. $P(205) = \frac{3075}{22} \approx 139.77$, so in 2008 ($205$ years after 1803), there were between 139 and 140 Sasquatch in Portage County.

\item \begin{enumerate}

\item $C(20) = 300$. It costs $\$300$ for 20 copies of the book.

\item $C(50) = 675$, so it costs $\$ 675$ for 50 copies of the book. $C(51) = 612$, so it costs $\$ 612$ for 51 copies of the book.

\item $56$ books.

\end{enumerate}

\item \begin{enumerate}

\item $S(10) = 17.5$, so it costs $\$ 17.50$ to ship 10 comic books.

\item There is free shipping on orders of $15$ or more comic books.

\end{enumerate}

\item \begin{enumerate}

\item $C(750) = 25$, so it costs $\$ 25$ to talk 750 minutes per month with this plan.

\item Since $20 \, \text{hours} = 1200 \, \text{minutes}$, we substitute $m = 1200$ and get $C(1200) = 45$. It costs $\$ 45$ to talk 20 hours per month with this plan.

\item It costs $\$25$ for up to $1000$ minutes and $10$ cents per minute for each minute over $1000$ minutes.

\end{enumerate}

\item \begin{enumerate}

\item $\lfloor 0.785 \rfloor = 0$, $\lfloor 117 \rfloor = 117$, $\lfloor -2.001 \rfloor = -3$, and $\lfloor \pi + 6 \rfloor = 9$

\end{enumerate}

\end{enumerate}

\closegraphsfile

1.5: Function Arithmetic

In Exercises \ref{basicarithonefirst} - \ref{basicarithonelast}, use the pair of functions $f$ and $g$ to find the following values if they exist.

\begin{multicols}{3}

\begin{itemize}

\item $(f+g)(2)$

\item $(f-g)(-1)$

\item $(g-f)(1)$

\end{itemize}

\end{multicols}

\begin{multicols}{3}

\begin{itemize}

\item $(fg)\left(\frac{1}{2}\right)$

\item $\left(\frac{f}{g}\right)(0)$

\item $\left(\frac{g}{f}\right)\left(-2\right)$

\end{itemize}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\item $f(x) = 3x+1$ and $g(x) = 4-x$ \label{basicarithonefirst}

\item $f(x) = x^2$ and $g(x) = -2x+1$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $f(x) = x^2 - x$ and $g(x) = 12-x^2$

\item $f(x) = 2x^3$ and $g(x) = -x^2-2x-3$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $f(x) = \sqrt{x+3}$ and $g(x) = 2x-1$

\item $f(x) = \sqrt{4-x}$ and $g(x) = \sqrt{x+2}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $f(x) = 2x$ and $g(x) = \dfrac{1}{2x+1}$

\item $f(x) = x^2$ and $g(x) = \dfrac{3}{2x-3}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $f(x) = x^2$ and $g(x) = \dfrac{1}{x^2}$

\item $f(x) = x^2+1$ and $g(x) = \dfrac{1}{x^2+1}$ \label{basicarithonelast}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

In Exercises \ref{basicarithtwofirst} - \ref{basicarithtwolast}, use the pair of functions $f$ and $g$ to find the domain of the indicated function then find and simplify an expression for it.

\begin{multicols}{4}

\begin{itemize}

\item $(f+g)(x)$

\item $(f-g)(x)$

\item $(fg)(x)$

\item $\left(\frac{f}{g}\right)(x)$

\end{itemize}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $f(x) = 2x+1$ and $g(x) = x-2$ \label{basicarithtwofirst}

\item $f(x) = 1-4x$ and $g(x) = 2x-1$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $f(x) = x^2$ and $g(x) = 3x-1$

\item $f(x) = x^2-x$ and $g(x) = 7x$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $f(x) = x^2-4$ and $g(x) = 3x+6$

\item $f(x) = -x^2+x+6$ and $g(x) = x^2-9$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $f(x) = \dfrac{x}{2}$ and $g(x) = \dfrac{2}{x}$

\item $f(x) =x-1$ and $g(x) = \dfrac{1}{x-1}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $f(x) = x$ and $g(x) = \sqrt{x+1}$

\item $f(x) =\sqrt{x-5}$ and $g(x) = f(x) = \sqrt{x-5}$ \label{basicarithtwolast}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

In Exercises \ref{diffquotexerfirst} - \ref{diffquotexerlast}, find and simplify the difference quotient $\dfrac{f(x+h) - f(x)}{h}$ for the given function.

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $f(x) = 2x - 5$ \label{diffquotexerfirst}

\item $f(x) = -3x + 5$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $f(x) = 6$

\item $f(x) = 3x^2 - x$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $f(x) = -x^2 + 2x - 1$

\item $f(x) = 4x^2$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $f(x) = x-x^2$

\item $f(x) = x^{3} + 1$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $f(x) = mx + b\;$ where $m \neq 0$

\item $f(x) = ax^{2} + bx + c\;$ where $a \neq 0$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $f(x) = \dfrac{2}{x}$

\item $f(x) = \dfrac{3}{1-x}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $f(x) = \dfrac{1}{x^2}$

\item $f(x) = \dfrac{2}{x+5}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $f(x) = \dfrac{1}{4x-3}$

\item $f(x) = \dfrac{3x}{x+1}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $f(x) = \dfrac{x}{x - 9}$

\item $f(x) = \dfrac{x^2}{2x+1}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $f(x) = \sqrt{x-9}$

\item $f(x) = \sqrt{2x+1}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $f(x) = \sqrt{-4x+5}$

\item $f(x) = \sqrt{4-x}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $f(x) = \sqrt{ax+b}$, where $a \neq 0$.

\item $f(x) = x \sqrt{x}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $f(x) = \sqrt[3]{x}$. \textbf{HINT:} $(a-b)\left(a^2+ab+b^2\right) = a^3 - b^3$ \label{diffquotexerlast}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

In Exercises \ref{econexerfirst} - \ref{econexerlast}, $C(x)$ denotes the cost to produce $x$ items and $p(x)$ denotes the price-demand function in the given economic scenario. In each Exercise, do the following:

\begin{multicols}{2}

\begin{itemize}

\item Find and interpret $C(0)$.

\item Find and interpret $\overline{C}(10)$.

\end{itemize}

\end{multicols}

\begin{multicols}{2}

\begin{itemize}

\item Find and interpret $p(5)$

\item Find and simplify $R(x)$.

\end{itemize}

\end{multicols}

\begin{multicols}{2}

\begin{itemize}

\item Find and simplify $P(x)$.

\item Solve $P(x) = 0$ and interpret.

\end{itemize}

\end{multicols}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item The cost, in dollars, to produce $x$ ``I'd rather be a Sasquatch'' T-Shirts is $C(x) = 2x+26$, $x \geq 0$ and the price-demand function, in dollars per shirt, is $p(x) = 30 - 2x$, $0 \leq x \leq 15$. \label{econexerfirst}

\item The cost, in dollars, to produce $x$ bottles of $100 \%$ All-Natural Certified Free-Trade Organic Sasquatch Tonic is $C(x) = 10x+100$, $x \geq 0$ and the price-demand function, in dollars per bottle, is $p(x) = 35 - x$, $0 \leq x \leq 35$.

\item The cost, in cents, to produce $x$ cups of Mountain Thunder Lemonade at Junior's Lemonade Stand is $C(x) = 18x + 240$, $x \geq 0$ and the price-demand function, in cents per cup, is $p(x) = 90-3x$, $0 \leq x \leq 30$.

\item The daily cost, in dollars, to produce $x$ Sasquatch Berry Pies $C(x) = 3x + 36$, $x \geq 0$ and the price-demand function, in dollars per pie, is $p(x) = 12-0.5x$, $0 \leq x \leq 24$.

\item The monthly cost, in hundreds of dollars, to produce $x$ custom built electric scooters is $C(x) = 20x + 1000$, $x \geq 0$ and the price-demand function, in hundreds of dollars per scooter, is $p(x) = 140-2x$, $0 \leq x \leq 70$. \label{econexerlast}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

In Exercises \ref{reformarithfirst} - \ref{reformarithlast}, let $f$ be the function defined by \[f = \{(-3, 4), (-2, 2), (-1, 0), (0, 1), (1, 3), (2, 4), (3, -1)\}\] and let $g$ be the function defined \[g = \{(-3, -2), (-2, 0), (-1, -4), (0, 0), (1, -3), (2, 1), (3, 2)\}\]. Compute the indicated value if it exists.

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $(f + g)(-3)$ \label{reformarithfirst}

\item $(f - g)(2)$

\item $(fg)(-1)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $(g + f)(1)$

\item $(g - f)(3)$

\item $(gf)(-3)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\left(\frac{f}{g}\right)(-2)$

\item $\left(\frac{f}{g}\right)(-1)$

\item $\left(\frac{f}{g}\right)(2)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\left(\frac{g}{f}\right)(-1)$

\item $\left(\frac{g}{f}\right)(3)$

\item $\left(\frac{g}{f}\right)(-3)$ \label{reformarithlast}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\newpage

\subsection{Answers}

\begin{enumerate}

\item For $f(x) = 3x+1$ and $g(x) = 4-x$

\begin{multicols}{3}

\begin{itemize}

\item $(f+g)(2) = 9$

\item $(f-g)(-1) = -7$

\item $(g-f)(1) = -1$

\end{itemize}

\end{multicols}

\begin{multicols}{3}

\begin{itemize}

\item $(fg)\left(\frac{1}{2}\right) = \frac{35}{4}$

\item $\left(\frac{f}{g}\right)(0) = \frac{1}{4}$

\item $\left(\frac{g}{f}\right)\left(-2\right) = -\frac{6}{5}$

\end{itemize}

\end{multicols}

\item For $f(x) = x^2$ and $g(x) = -2x+1$

\begin{multicols}{3}

\begin{itemize}

\item $(f+g)(2) = 1$

\item $(f-g)(-1) = -2$

\item $(g-f)(1) = -2$

\end{itemize}

\end{multicols}

\begin{multicols}{3}

\begin{itemize}

\item $(fg)\left(\frac{1}{2}\right) = 0$

\item $\left(\frac{f}{g}\right)(0) = 0$

\item $\left(\frac{g}{f}\right)\left(-2\right) = \frac{5}{4}$

\end{itemize}

\end{multicols}

\item For $f(x) = x^2 - x$ and $g(x) = 12-x^2$

\begin{multicols}{3}

\begin{itemize}

\item $(f+g)(2) = 10$

\item $(f-g)(-1) = -9$

\item $(g-f)(1) = 11$

\end{itemize}

\end{multicols}

\begin{multicols}{3}

\begin{itemize}

\item $(fg)\left(\frac{1}{2}\right) = -\frac{47}{16}$

\item $\left(\frac{f}{g}\right)(0) = 0$

\item $\left(\frac{g}{f}\right)\left(-2\right) = \frac{4}{3}$

\end{itemize}

\end{multicols}

\item For $f(x) = 2x^3$ and $g(x) = -x^2-2x-3$

\begin{multicols}{3}

\begin{itemize}

\item $(f+g)(2) = 5$

\item $(f-g)(-1) = 0$

\item $(g-f)(1) = -8$

\end{itemize}

\end{multicols}

\begin{multicols}{3}

\begin{itemize}

\item $(fg)\left(\frac{1}{2}\right) = -\frac{17}{16}$

\item $\left(\frac{f}{g}\right)(0) = 0$

\item $\left(\frac{g}{f}\right)\left(-2\right) = \frac{3}{16}$

\end{itemize}

\end{multicols}

\item For $f(x) = \sqrt{x+3}$ and $g(x) = 2x-1$

\begin{multicols}{3}

\begin{itemize}

\item $(f+g)(2) = 3+\sqrt{5}$

\item $(f-g)(-1) = 3+\sqrt{2}$

\item $(g-f)(1) = -1$

\end{itemize}

\end{multicols}

\begin{multicols}{3}

\begin{itemize}

\item $(fg)\left(\frac{1}{2}\right) = 0$

\item $\left(\frac{f}{g}\right)(0) = -\sqrt{3}$

\item $\left(\frac{g}{f}\right)\left(-2\right) = -5$

\end{itemize}

\end{multicols}

\item For $f(x) = \sqrt{4-x}$ and $g(x) = \sqrt{x+2}$

\begin{multicols}{3}

\begin{itemize}

\item $(f+g)(2) = 2+\sqrt{2}$

\item $(f-g)(-1) = -1+\sqrt{5}$

\item $(g-f)(1) = 0$

\end{itemize}

\end{multicols}

\begin{multicols}{3}

\begin{itemize}

\item $(fg)\left(\frac{1}{2}\right) = \frac{\sqrt{35}}{2}$

\item $\left(\frac{f}{g}\right)(0) = \sqrt{2}$

\item $\left(\frac{g}{f}\right)\left(-2\right) = 0$

\end{itemize}

\end{multicols}

\newpage

\item For $f(x) = 2x$ and $g(x) = \frac{1}{2x+1}$

\begin{multicols}{3}

\begin{itemize}

\item $(f+g)(2) = \frac{21}{5}$

\item $(f-g)(-1) = -1$

\item $(g-f)(1) = -\frac{5}{3}$

\end{itemize}

\end{multicols}

\begin{multicols}{3}

\begin{itemize}

\item $(fg)\left(\frac{1}{2}\right) = \frac{1}{2}$

\item $\left(\frac{f}{g}\right)(0) = 0$

\item $\left(\frac{g}{f}\right)\left(-2\right) = \frac{1}{12}$

\end{itemize}

\end{multicols}

\item For $f(x) = x^2$ and $g(x) = \frac{3}{2x-3}$

\begin{multicols}{3}

\begin{itemize}

\item $(f+g)(2) = 7$

\item $(f-g)(-1) = \frac{8}{5}$

\item $(g-f)(1) = -4$

\end{itemize}

\end{multicols}

\begin{multicols}{3}

\begin{itemize}

\item $(fg)\left(\frac{1}{2}\right) = -\frac{3}{8}$

\item $\left(\frac{f}{g}\right)(0) = 0$

\item $\left(\frac{g}{f}\right)\left(-2\right) = -\frac{3}{28}$

\end{itemize}

\end{multicols}

\item For $f(x) = x^2$ and $g(x) = \frac{1}{x^2}$

\begin{multicols}{3}

\begin{itemize}

\item $(f+g)(2) =\frac{17}{4}$

\item $(f-g)(-1) = 0$

\item $(g-f)(1) = 0$

\end{itemize}

\end{multicols}

\begin{multicols}{3}

\begin{itemize}

\item $(fg)\left(\frac{1}{2}\right) =1$

\item $\left(\frac{f}{g}\right)(0)$ is undefined.

\item $\left(\frac{g}{f}\right)\left(-2\right) = \frac{1}{16}$

\end{itemize}

\end{multicols}

\item For $f(x) = x^2+1$ and $g(x) = \frac{1}{x^2+1}$

\begin{multicols}{3}

\begin{itemize}

\item $(f+g)(2) =\frac{26}{5}$

\item $(f-g)(-1) = \frac{3}{2}$

\item $(g-f)(1) = -\frac{3}{2}$

\end{itemize}

\end{multicols}

\begin{multicols}{3}

\begin{itemize}

\item $(fg)\left(\frac{1}{2}\right) =1$

\item $\left(\frac{f}{g}\right)(0) = 1$

\item $\left(\frac{g}{f}\right)\left(-2\right) = \frac{1}{25}$

\end{itemize}

\end{multicols}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item For $f(x) = 2x+1$ and $g(x) = x-2$

\begin{multicols}{2}

\begin{itemize}

\item $(f+g)(x) = 3x-1$ \\

Domain: $(-\infty, \infty)$

\vfill

\columnbreak

\item $(f-g)(x) = x+3$ \\

Domain: $(-\infty, \infty)$

\end{itemize}

\end{multicols}

\begin{multicols}{2}

\begin{itemize}

\item $(fg)(x) = 2x^2-3x-2$ \\

Domain: $(-\infty, \infty)$

\vfill

\columnbreak

\item $\left(\frac{f}{g}\right)(x) = \frac{2x+1}{x-2}$ \\

Domain: $(-\infty, 2) \cup (2, \infty)$

\end{itemize}

\end{multicols}

\item For $f(x) = 1-4x$ and $g(x) = 2x-1$

\begin{multicols}{2}

\begin{itemize}

\item $(f+g)(x) = -2x$ \\

Domain: $(-\infty, \infty)$

\vfill

\columnbreak

\item $(f-g)(x) = 2-6x$ \\

Domain: $(-\infty, \infty)$

\end{itemize}

\end{multicols}

\begin{multicols}{2}

\begin{itemize}

\item $(fg)(x) = -8x^2+6x-1$ \\

Domain: $(-\infty, \infty)$

\vfill

\columnbreak

\item $\left(\frac{f}{g}\right)(x) = \frac{1-4x}{2x-1}$ \\

Domain: $\left(-\infty, \frac{1}{2} \right) \cup \left(\frac{1}{2}, \infty \right)$

\end{itemize}

\end{multicols}

\newpage

\item For $f(x) = x^2$ and $g(x) = 3x-1$

\begin{multicols}{2}

\begin{itemize}

\item $(f+g)(x) = x^2+3x-1$ \\

Domain: $(-\infty, \infty)$

\vfill

\columnbreak

\item $(f-g)(x) = x^2-3x+1$ \\

Domain: $(-\infty, \infty)$

\end{itemize}

\end{multicols}

\begin{multicols}{2}

\begin{itemize}

\item $(fg)(x) = 3x^3-x^2$ \\

Domain: $(-\infty, \infty)$

\vfill

\columnbreak

\item $\left(\frac{f}{g}\right)(x) = \frac{x^2}{3x-1}$ \\

Domain: $\left(-\infty, \frac{1}{3} \right) \cup \left(\frac{1}{3}, \infty \right)$

\end{itemize}

\end{multicols}

\item For $f(x) = x^2-x$ and $g(x) = 7x$

\begin{multicols}{2}

\begin{itemize}

\item $(f+g)(x) = x^2+6x$ \\

Domain: $(-\infty, \infty)$

\vfill

\columnbreak

\item $(f-g)(x) = x^2-8x$ \\

Domain: $(-\infty, \infty)$

\end{itemize}

\end{multicols}

\begin{multicols}{2}

\begin{itemize}

\item $(fg)(x) = 7x^3-7x^2$ \\

Domain: $(-\infty, \infty)$

\vfill

\columnbreak

\item $\left(\frac{f}{g}\right)(x) = \frac{x-1}{7}$ \\

Domain: $\left(-\infty, 0 \right) \cup \left(0, \infty \right)$

\end{itemize}

\end{multicols}

\item For $f(x) = x^2-4$ and $g(x) = 3x+6$

\begin{multicols}{2}

\begin{itemize}

\item $(f+g)(x) = x^2+3x+2$ \\

Domain: $(-\infty, \infty)$

\vfill

\columnbreak

\item $(f-g)(x) = x^2-3x-10$ \\

Domain: $(-\infty, \infty)$

\end{itemize}

\end{multicols}

\begin{multicols}{2}

\begin{itemize}

\item $(fg)(x) = 3x^3+6x^2-12x-24$ \\

Domain: $(-\infty, \infty)$

\vfill

\columnbreak

\item $\left(\frac{f}{g}\right)(x) = \frac{x-2}{3}$ \\

Domain: $\left(-\infty, -2 \right) \cup \left(-2, \infty \right)$

\end{itemize}

\end{multicols}

\item For $f(x) = -x^2+x+6$ and $g(x) = x^2-9$

\begin{multicols}{2}

\begin{itemize}

\item $(f+g)(x) = x-3$ \\

Domain: $(-\infty, \infty)$

\vfill

\columnbreak

\item $(f-g)(x) = -2x^2+x+15$ \\

Domain: $(-\infty, \infty)$

\end{itemize}

\end{multicols}

\begin{multicols}{2}

\begin{itemize}

\item $(fg)(x) = -x^4+x^3+15x^2-9x-54$ \\

Domain: $(-\infty, \infty)$

\vfill

\columnbreak

\item $\left(\frac{f}{g}\right)(x) = -\frac{x+2}{x+3}$ \\

Domain: $\left(-\infty, -3 \right) \cup \left(-3, 3 \right) \cup (3, \infty)$

\end{itemize}

\end{multicols}

\item For $f(x) = \frac{x}{2}$ and $g(x) = \frac{2}{x}$

\begin{multicols}{2}

\begin{itemize}

\item $(f+g)(x) = \frac{x^2+4}{2x}$ \\

Domain: $(-\infty, 0) \cup (0, \infty)$

\vfill

\columnbreak

\item $(f-g)(x) = \frac{x^2-4}{2x}$ \\

Domain: $(-\infty,0) \cup (0, \infty)$

\end{itemize}

\end{multicols}

\begin{multicols}{2}

\begin{itemize}

\item $(fg)(x) = 1$ \\

Domain: $(-\infty,0) \cup (0, \infty)$

\vfill

\columnbreak

\item $\left(\frac{f}{g}\right)(x) = \frac{x^2}{4}$ \\

Domain: $(-\infty,0) \cup (0, \infty)$

\end{itemize}

\end{multicols}

\newpage

\item For $f(x) =x-1$ and $g(x) = \frac{1}{x-1}$

\begin{multicols}{2}

\begin{itemize}

\item $(f+g)(x) = \frac{x^2-2x+2}{x-1}$ \\

Domain: $(-\infty, 1) \cup (1, \infty)$

\vfill

\columnbreak

\item $(f-g)(x) = \frac{x^2-2x}{x-1}$ \\

Domain: $(-\infty,1) \cup (1, \infty)$

\end{itemize}

\end{multicols}

\begin{multicols}{2}

\begin{itemize}

\item $(fg)(x) = 1$ \\

Domain: $(-\infty,1) \cup (1, \infty)$

\vfill

\columnbreak

\item $\left(\frac{f}{g}\right)(x) =x^2-2x+1$ \\

Domain: $(-\infty,1) \cup (1, \infty)$

\end{itemize}

\end{multicols}

\item For $f(x) =x$ and $g(x) = \sqrt{x+1}$

\begin{multicols}{2}

\begin{itemize}

\item $(f+g)(x) = x+\sqrt{x+1}$ \\

Domain: $[-1,\infty)$

\vfill

\columnbreak

\item $(f-g)(x) = x-\sqrt{x+1}$ \\

Domain: $[-1,\infty)$

\end{itemize}

\end{multicols}

\begin{multicols}{2}

\begin{itemize}

\item $(fg)(x) = x\sqrt{x+1}$ \\

Domain: $[-1,\infty)$

\vfill

\columnbreak

\item $\left(\frac{f}{g}\right)(x) =\frac{x}{\sqrt{x+1}}$ \\

Domain: $(-1,\infty)$

\end{itemize}

\end{multicols}

\item For $f(x) = \sqrt{x-5}$ and $g(x) = f(x) = \sqrt{x-5}$

\begin{multicols}{2}

\begin{itemize}

\item $(f+g)(x) = 2\sqrt{x-5}$ \\

Domain: $[5,\infty)$

\vfill

\columnbreak

\item $(f-g)(x) =0$ \\

Domain: $[5,\infty)$

\end{itemize}

\end{multicols}

\begin{multicols}{2}

\begin{itemize}

\item $(fg)(x) =x-5$ \\

Domain: $[5,\infty)$

\vfill

\columnbreak

\item $\left(\frac{f}{g}\right)(x) =1$ \\

Domain: $(5,\infty)$

\end{itemize}

\end{multicols}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $2$

\item $-3$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $0$

\item $6x+3h-1$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $-2x-h+2$

\item $8x+4h$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $-2x-h+1$

\item $3x^{2} + 3xh + h^{2}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $m$

\item $2ax + ah + b$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\dfrac{-2}{x(x+h)}$

\item $\dfrac{3}{(1-x-h)(1-x)}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\dfrac{-(2x+h)}{x^2(x+h)^2}$

\item $\dfrac{-2}{(x+5)(x+h+5)}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\dfrac{-4}{(4x-3)(4x+4h-3)}$

\item $\dfrac{3}{(x+1)(x+h+1)}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\dfrac{-9}{(x - 9)(x + h - 9)}$

\item $\dfrac{2x^2+2xh+2x+h}{(2x+1)(2x+2h+1)}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\dfrac{1}{\sqrt{x+h-9} + \sqrt{x-9}}$

\item $\dfrac{2}{\sqrt{2x+2h+1} + \sqrt{2x+1}}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\dfrac{-4}{\sqrt{-4x-4h+5} + \sqrt{-4x+5}}$

\item $\dfrac{-1}{\sqrt{4-x-h} + \sqrt{4-x}}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\dfrac{a}{\sqrt{ax+ah+b} + \sqrt{ax+b}}$

\item $\dfrac{3x^2+3xh+h^2}{(x+h)^{3/2} + x^{3/2}} $

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\dfrac{1}{(x+h)^{2/3} + (x+h)^{1/3} x^{1/3} + x^{2/3}}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item \begin{itemize}

\item $C(0) = 26$, so the fixed costs are $\$26$.

\item $\overline{C}(10) = 4.6$, so when 10 shirts are produced, the cost per shirt is $\$4.60$.

\item $p(5) = 20$, so to sell $5$ shirts, set the price at $\$20$ per shirt.

\item $R(x) = -2x^2+30x$, $0 \leq x \leq 15$

\item $P(x) = -2x^2+28x-26$, $0 \leq x \leq 15$

\item $P(x) = 0$ when $x = 1$ and $x=13$. These are the `break even' points, so selling $1$ shirt or $13$ shirts will guarantee the revenue earned exactly recoups the cost of production.

\end{itemize}

\item \begin{itemize}

\item $C(0) = 100$, so the fixed costs are $\$100$.

\item $\overline{C}(10) = 20$, so when 10 bottles of tonic are produced, the cost per bottle is $\$20$.

\item $p(5) = 30$, so to sell $5$ bottles of tonic, set the price at $\$30$ per bottle.

\item $R(x) = -x^2+35x$, $0 \leq x \leq 35$

\item $P(x) = -x^2+25x-100$, $0 \leq x \leq 35$

\item $P(x) = 0$ when $x = 5$ and $x=20$. These are the `break even' points, so selling $5$ bottles of tonic or $20$ bottles of tonic will guarantee the revenue earned exactly recoups the cost of production.

\end{itemize}

\item \begin{itemize}

\item $C(0) = 240$, so the fixed costs are $240$\textcent \, or $\$2.40$.

\item $\overline{C}(10) = 42$, so when 10 cups of lemonade are made, the cost per cup is $42$\textcent.

\item $p(5) = 75$, so to sell $5$ cups of lemonade, set the price at $75$\textcent \, per cup.

\item $R(x) = -3x^2+90x$, $0 \leq x \leq 30$

\item $P(x) = -3x^2+72x-240$, $0 \leq x \leq 30$

\item $P(x) = 0$ when $x = 4$ and $x=20$. These are the `break even' points, so selling $4$ cups of lemonade or $20$ cups of lemonade will guarantee the revenue earned exactly recoups the cost of production.

\end{itemize}

\pagebreak

\item \begin{itemize}

\item $C(0) = 36$, so the daily fixed costs are $\$36$.

\item $\overline{C}(10) = 6.6$, so when 10 pies are made, the cost per pie is $\$6.60$.

\item $p(5) = 9.5$, so to sell $5$ pies a day, set the price at $\$9.50$ per pie.

\item $R(x) = -0.5 x^2 + 12x$, $0 \leq x \leq 24$

\item $P(x) = -0.5 x^2+9x-36$, $0 \leq x \leq 24$

\item $P(x) = 0$ when $x = 6$ and $x=12$. These are the `break even' points, so selling $6$ pies or $12$ pies a day will guarantee the revenue earned exactly recoups the cost of production.

\end{itemize}

\item \begin{itemize}

\item $C(0) = 1000$, so the monthly fixed costs are $1000$ \textit{hundred} dollars, or $\$100,\!000$.

\item $\overline{C}(10) = 120$, so when 10 scooters are made, the cost per scooter is $120$ hundred dollars, or $\$12,\!000$.

\item $p(5) = 130$, so to sell $5$ scooters a month, set the price at $130$ hundred dollars, or $\$13,\!000$ per scooter.

\item $R(x) = -2x^2+140x$, $0 \leq x \leq 70$

\item $P(x) = -2x^2+120x-1000$, $0 \leq x \leq 70$

\item $P(x) = 0$ when $x = 10$ and $x=50$. These are the `break even' points, so selling $10$ scooters or $50$ scooters a month will guarantee the revenue earned exactly recoups the cost of production.

\end{itemize}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $(f + g)(-3) = 2$

\item $(f - g)(2) = 3$

\item $(fg)(-1) = 0$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $(g + f)(1) = 0$

\item $(g - f)(3) = 3$

\item $(gf)(-3) = -8$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\left(\frac{f}{g}\right)(-2)$ does not exist

\item $\left(\frac{f}{g}\right)(-1) = 0$

\item $\left(\frac{f}{g}\right)(2) = 4$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\left(\frac{g}{f}\right)(-1)$ does not exist

\item $\left(\frac{g}{f}\right)(3) = -2$

\item $\left(\frac{g}{f}\right)(-3) = -\frac{1}{2}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\closegraphsfile

1.6: Graphs of Functions

\subsection{Exercises}

\begin{enumerate}

\item For each equation given below

\begin{itemize}

\item Find the $x$- and $y$-intercept(s) of the graph, if any exist.

\item Following the procedure in Example \hspace{-.1in} ~\ref{firstequgraph}, create a table of sample points on the graph of the equation.

\item Plot the sample points and create a rough sketch of the graph of the equation.

\item Test for symmetry. If the equation appears to fail any of the symmetry tests, find a point on the graph of the equation whose reflection fails to be on the graph as was done at the end of Example \hspace{-.1in} ~\ref{secondequgraph}

\end{itemize}

\begin{multicols}{2}

\begin{enumerate}

\item $y = x^{2} + 1$

\item $y = x^2-2x-8$

\item $y = x^{3} - x$

\item $y = \frac{x^3}{4} - 3x$

\item $y = \sqrt{x - 2}$

\item $y = 2 \sqrt{x+4} - 2$

\item $3x - y = 7$

\item $3x-2y = 10$

\item $(x+2)^2+y^2 = 16$

\item $x^{2} - y^{2} = 1$

\item $4y^2 - 9x^2 = 36$

\item $x^{3}y = -4$

\end{enumerate}

\end{multicols}

\item The procedures which we have outlined in the Examples of this section and used in the exercises given above all rely on the fact that the equations were ``well-behaved''. Not everything in Mathematics is quite so tame, as the following equations will show you. Discuss with your classmates how you might approach graphing these equations. What difficulties arise when trying to apply the various tests and procedures given in this section? For more information, including pictures of the curves, each curve name is a link to its page at www.wikipedia.org. For a much longer list of fascinating curves, click \href{http://en.wikipedia.org/wiki/List_of...derline{here}}.

\label{listofcurves}

\begin{multicols}{2}

\begin{enumerate}

\item $x^{3} + y^{3} - 3xy = 0\;$ \href{http://en.wikipedia.org/wiki/Folium_...derline{Folium of Descartes}}

\item $x^{4} = x^{2} + y^{2}\;$ \href{http://en.wikipedia.org/wiki/Kampyle...erline{Kampyle of Eudoxus}}

\item $y^{2} = x^{3} + 3x^{2}\;$ \href{http://en.wikipedia.org/wiki/Tschirn...{Tschirnhausen cubic}}

\item $(x^{2} + y^{2})^{2} = x^{3} + y^{3}\;$ \href{http://en.wikipedia.org/wiki/Crooked...erline{Crooked egg}}

\end{enumerate}

\end{multicols}

\end{enumerate}

\newpage

\subsection{Answers}

\begin{enumerate}

\item \begin{multicols}{2} \raggedcolumns

\begin{enumerate}

\item $y = x^{2} + 1$

\begin{flushleft}

The graph has no $x$-intercepts

$y$-intercept: $(0, 1)$

$\begin{array}{|r||c|c|}

\hline

x & y & (x,y) \\ \hline

-2 & 5 & (-2, 5) \\ \hline

-1 & 2 & (-1, 2) \\ \hline

0 & 1 & (0, 1) \\ \hline

1 & 2 & (1, 2) \\ \hline

2 & 5 & (2, 5) \\ \hline

\end{array} $

\begin{mfpic}[10]{-3}{3}{-1}{6}

\point[3pt]{(-2,5), (-1,2), (0,1), (1,2), (2,5)}

\axes

\tlabel[cc](3,-0.5){\scriptsize $x$}

\tlabel[cc](0.5,6){\scriptsize $y$}

\xmarks{-2,-1,1,2}

\ymarks{1,2,3,4,5}

\tlpointsep{4pt}

\axislabels {x}{{\tiny $-2 \hspace{6pt}$} -2, {\tiny $-1 \hspace{6pt}$} -1, {\tiny $1$} 1, {\tiny $2$} 2}

\axislabels {y}{{\tiny $1$} 1, {\tiny $2$} 2, {\tiny $3$} 3, {\tiny $4$} 4, {\tiny $5$} 5}

\arrow \reverse \arrow \function{-2.3, 2.3, 0.1}{x**2+1}

\end{mfpic}

The graph is not symmetric about the $x$-axis (e.g. $(2, 5)$ is on the graph but $(2, -5)$ is not)

The graph is symmetric about the $y$-axis

The graph is not symmetric about the origin (e.g. $(2, 5)$ is on the graph but $(-2, -5)$ is not)

\end{flushleft}

\vspace{4in}

\item $y = x^{2} -2x-8$

\begin{flushleft}

$x$-intercepts: $(4,0)$, $(-2,0)$

$y$-intercept: $(0, -8)$

$\begin{array}{|r||c|c|}

\hline

x & y & (x,y) \\ \hline

-3 & 7 & (-3,7) \\ \hline

-2 & 0 & (-2, 0) \\ \hline

-1 & -5 & (-1, -5) \\ \hline

0 & -8 & (0, -8) \\ \hline

1 & -9 & (1, -9) \\ \hline

2 & -8 & (2, -8) \\ \hline

3 & -5 & (3,-5) \\ \hline

4 & 0 & (4,0) \\ \hline

5 & 7 & (5,7) \\ \hline

\end{array}$

\begin{mfpic}[7]{-4}{6}{-10}{8}

\point[3pt]{(-3,7), (-2,0), (-1,-5), (0,-8), (1,-9), (2,-8), (3,-5), (4,0), (5,7)}

\axes

\tlabel[cc](6,-0.5){\scriptsize $x$}

\tlabel[cc](0.5,8){\scriptsize $y$}

\xmarks{-3,-2,-1,1,2,3,4,5}

\ymarks{-9,-8,-7,-6,-5,-4,-3,-2,-1,1,2,3,4,5,6,7}

\tlpointsep{4pt}

\axislabels {x}{{\tiny $-3 \hspace{6pt}$} -3,{\tiny $-2 \hspace{6pt}$} -2, {\tiny $-1 \hspace{6pt}$} -1, {\tiny $1$} 1, {\tiny $2$} 2, {\tiny $3$} 3, {\tiny $4$} 4, {\tiny $5$} 5}

\axislabels {y}{{\tiny $-9$} -9, {\tiny $-8$} -8, {\tiny $-7$} -7, {\tiny $-6$} -6, {\tiny $-5$} -5, {\tiny $-4$} -4, {\tiny $-3$} -3, {\tiny $-2$} -2, {\tiny $1$} 1, {\tiny $2$} 2, {\tiny $3$} 3, {\tiny $4$} 4, {\tiny $5$} 5, {\tiny $6$} 6, {\tiny $7$} 7}

\arrow \reverse \arrow \function{-3.1, 5.1, 0.1}{x**2-2*x-8}

\end{mfpic}

The graph is not symmetric about the $x$-axis (e.g. $(-3, 7)$ is on the graph but $(-3, -7)$ is not)

The graph is not symmetric about the $y$-axis (e.g. $(-3, 7)$ is on the graph but $(3, 7)$ is not)

The graph is not symmetric about the origin (e.g. $(-3, 7)$ is on the graph but $(3, -7)$ is not)

\end{flushleft}

\pagebreak

\item $y = x^{3} - x$

\begin{flushleft}

$x$-intercepts: $(-1, 0), (0, 0), (1, 0)$

$y$-intercept: $(0, 0)$

$\begin{array}{|r||c|c|}

\hline

x & y & (x,y) \\ \hline

-2 & -6 & (-2, -6) \\ \hline

-1 & 0 & (-1, 0) \\ \hline

0 & 0 & (0, 0) \\ \hline

1 & 0 & (1, 0) \\ \hline

2 & 6 & (2, 6) \\ \hline

\end{array} $

\begin{mfpic}[10]{-3}{3}{-7}{7}

\point[3pt]{(-2,-6), (-1,0), (0,0), (1,0), (2,6)}

\axes

\tlabel[cc](3,-0.5){\scriptsize $x$}

\tlabel[cc](0.5,7){\scriptsize $y$}

\xmarks{-2,-1,1,2}

\ymarks{-6,-5,-4,-3,-2,-1,1,2,3,4,5,6}

\tlpointsep{4pt}

\axislabels {x}{{\tiny $-2 \hspace{6pt}$} -2, {\tiny $-1 \hspace{6pt}$} -1, {\tiny $1$} 1, {\tiny $2$} 2}

\axislabels {y}{{\tiny $-6$} -6,{\tiny $-5$} -5,{\tiny $-4$} -4,{\tiny $-3$} -3,{\tiny $-2$} -2,{\tiny $-1$} -1, {\tiny $1$} 1, {\tiny $2$} 2, {\tiny $3$} 3, {\tiny $4$} 4, {\tiny $5$} 5, {\tiny $6$} 6}

\arrow \reverse \arrow \function{-2.1, 2.1, 0.1}{x**3-x}

\end{mfpic}

The graph is not symmetric about the $x$-axis. (e.g. $(2, 6)$ is on the graph but $(2, -6)$ is not)

The graph is not symmetric about the $y$-axis. (e.g. $(2, 6)$ is on the graph but $(-2, 6)$ is not)

The graph is symmetric about the origin.

\end{flushleft}

\vspace{2in}

\item $y = \frac{x^3}{4} - 3x$

\begin{flushleft}

$x$-intercepts: $\left(\pm 2\sqrt{3}, 0\right)$

$y$-intercept: $(0,0)$

$\begin{array}{|r||c|c|}

\hline

x & y & (x,y) \\ \hline

-4 & -4 & (-4, -4) \\ \hline

-3 & \frac{9}{4} & \left(-3, \frac{9}{4} \right) \\ \hline

-2 & 4 & (-2, 4) \\ \hline

-1 & \frac{11}{4} & \left(-1, \frac{11}{4}\right) \\ \hline

0 & 0 & (0,0) \\ \hline

1 & -\frac{11}{4} & \left(1, -\frac{11}{4}\right) \\ \hline

2 & -4 & (2, -4) \\ \hline

3 & -\frac{9}{4} & \left(3, -\frac{9}{4} \right) \\ \hline

4 & 4 & (4, 4) \\ \hline

\end{array} $

\begin{mfpic}[10]{-5}{5}{-5}{5}

\point[3pt]{(-4,-4), (-3.4641, 0), (-3, 2.25), (-2,4), (-1,2.75), (0,0), (4,4), (3.4641, 0),(3, -2.25), (2,-4), (1,-2.75)}

\axes

\tlabel[cc](5,-0.5){\scriptsize $x$}

\tlabel[cc](0.5,5){\scriptsize $y$}

\xmarks{-4,-3,-2,-1,1,2,3,4}

\ymarks{-4,-3,-2,-1,1,2,3,4}

\tlpointsep{4pt}

\axislabels {x}{{\tiny $-4 \hspace{6pt}$} -4,{\tiny $-3 \hspace{6pt}$} -3,{\tiny $-2 \hspace{6pt}$} -2,{\tiny $-1 \hspace{6pt}$} -1,{\tiny $1$} 1, {\tiny $2$} 2, {\tiny $3$} 3, {\tiny $4$} 4}

\axislabels {y}{{\tiny $-1$} -1, {\tiny $-2$} -2, {\tiny $-3$} -3, {\tiny $-4$} -4,{\tiny $1$} 1, {\tiny $2$} 2, {\tiny $3$} 3, {\tiny $4$} 4}

\arrow \reverse \arrow \function{-4.1, 4.1, 0.1}{0.25*(x**3)-3*x}

\end{mfpic}

The graph is not symmetric about the $x$-axis (e.g. $(-4, -4)$ is on the graph but $(-4, 4)$ is not)

The graph is not symmetric about the $y$-axis (e.g. $(-4, -4)$ is on the graph but $(4, -4)$ is not)

The graph is symmetric about the origin

\end{flushleft}

\pagebreak

\item $y = \sqrt{x - 2}$

\begin{flushleft}

$x$-intercept: $(2, 0)$

The graph has no $y$-intercepts

$\begin{array}{|r||c|c|}

\hline

x & y & (x,y) \\ \hline

2 & 0 & (2, 0) \\ \hline

3 & 1 & (3, 1) \\ \hline

6 & 2 & (6, 2) \\ \hline

11 & 3 & (11, 3) \\ \hline

\end{array} $

\begin{mfpic}[10]{-1}{12}{-1}{4}

\point[3pt]{(2,0), (3,1), (6,2), (11,3)}

\axes

\tlabel[cc](12,-0.5){\scriptsize $x$}

\tlabel[cc](0.5,4){\scriptsize $y$}

\xmarks{1,2,3,4,5,6,7,8,9,10,11}

\ymarks{1,2,3}

\tlpointsep{4pt}

\axislabels {x}{{\tiny $1$} 1, {\tiny $2$} 2, {\tiny $3$} 3, {\tiny $4$} 4, {\tiny $5$} 5, {\tiny $6$} 6, {\tiny $7$} 7, {\tiny $8$} 8, {\tiny $9$} 9, {\tiny $10$} 10, {\tiny $11$} 11}

\axislabels {y}{{\tiny $1$} 1, {\tiny $2$} 2, {\tiny $3$} 3}

\arrow \function{2, 12, 0.1}{sqrt(x - 2)}

\end{mfpic}

The graph is not symmetric about the $x$-axis (e.g. $(3, 1)$ is on the graph but $(3, -1)$ is not)

The graph is not symmetric about the $y$-axis (e.g. $(3, 1)$ is on the graph but $(-3, 1)$ is not)

The graph is not symmetric about the origin (e.g. $(3, 1)$ is on the graph but $(-3, -1)$ is not)

\end{flushleft}

\vspace{4in}

\item $y = 2 \sqrt{x+4} - 2$

\begin{flushleft}

$x$-intercept: $(-3,0)$

$y$-intercept: $(0,2)$

$\begin{array}{|r||c|c|}

\hline

x & y & (x,y) \\ \hline

-4 & -2 & (-4, -2) \\ \hline

-3 & 0 & (-3,0 ) \\ \hline

-2 & 2 \sqrt{2} -2 & \left(-2, \sqrt{2} -2 \right) \\ \hline

-1 & 2 \sqrt{3} -2 & \left(-2, \sqrt{3} -2 \right) \\ \hline

0 & 2 & (0, 2) \\ \hline

1 & 2 \sqrt{5} -2 & \left(-2, \sqrt{5} -2 \right) \\ \hline

\end{array} $

\begin{mfpic}[10]{-5}{3}{-4}{4}

\point[3pt]{(-4,-2), (-3,0), (-2, 0.8284), (-1, 1.464), (0,2), (1,2.472)}

\axes

\tlabel[cc](3,-0.5){\scriptsize $x$}

\tlabel[cc](0.5,4){\scriptsize $y$}

\xmarks{-4,-3,-2,-1,1,2}

\ymarks{-3,-2,-1,1,2,3}

\tlpointsep{4pt}

\axislabels {x}{{\tiny $-4 \hspace{6pt}$} -4,{\tiny $-3 \hspace{6pt}$} -3, {\tiny $-2 \hspace{6pt}$} -2, {\tiny $-1 \hspace{6pt}$} -1, {\tiny $1$} 1, {\tiny $2$} 2}

\axislabels {y}{{\tiny $-3$} -3, {\tiny $-2$} -2, {\tiny $-1$} -1, {\tiny $1$} 1, {\tiny $2$} 2, {\tiny $3$} 3}

\arrow \function{-4,2,0.1}{2 * sqrt(x+4)-2}

\end{mfpic}

The graph is not symmetric about the $x$-axis (e.g. $(-4, -2)$ is on the graph but $(-4, 2)$ is not)

The graph is not symmetric about the $y$-axis (e.g. $(-4, -2)$ is on the graph but $(4, -2)$ is not)

The graph is not symmetric about the origin (e.g. $(-4, -2)$ is on the graph but $(4, 2)$ is not)

\end{flushleft}

\pagebreak

\item $3x - y = 7$ \\ Re-write as: $y = 3x - 7$.

\begin{flushleft}

$x$-intercept: $(\frac{7}{3}, 0)$

$y$-intercept: $(0, -7)$

$\begin{array}{|r||c|c|}

\hline

x & y & (x,y) \\ \hline

-2 & -13 & (-2,-13) \\ \hline

-1 & -10 & (-1,-10) \\ \hline

0 & -7 & (0, -7) \\ \hline

1 & -4 & (1, -4) \\ \hline

2 & -1 & (2, -1) \\ \hline

3 & 2 & (3, 2) \\ \hline

\end{array} $

\begin{mfpic}[10]{-3}{4}{-14}{4}

\point[3pt]{(-2,-13), (-1,-10), (0, -7), (1, -4), (2, -1), (3, 2)}

\axes

\tlabel[cc](4,-0.5){\scriptsize $x$}

\tlabel[cc](0.5,4){\scriptsize $y$}

\xmarks{-2,-1,1,2,3}

\ymarks{-13,-12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1,1,2,3}

\tlpointsep{4pt}

\axislabels {x}{{\tiny $-2 \hspace{6pt}$} -2, {\tiny $-1 \hspace{6pt}$} -1, {\tiny $1$} 1, {\tiny $2$} 2, {\tiny $3$} 3}

\axislabels {y}{{\tiny $-13$} -13, {\tiny $-12$} -12, {\tiny $-11$} -11, {\tiny $-10$} -10, {\tiny $-9$} -9, {\tiny $-8$} -8, {\tiny $-7$} -7, {\tiny $-6$} -6, {\tiny $-5$} -5, {\tiny $-4$} -4, {\tiny $-3$} -3, {\tiny $-2$} -2, {\tiny $-1$} -1, {\tiny $1$} 1, {\tiny $2$} 2, {\tiny $3$} 3}

\arrow \reverse \arrow \function{-2.2, 3.2, 0.1}{3*x - 7}

\end{mfpic}

The graph is not symmetric about the $x$-axis (e.g. $(3, 2)$ is on the graph but $(3, -2)$ is not)

The graph is not symmetric about the $y$-axis (e.g. $(3, 2)$ is on the graph but $(-3, 2)$ is not)

The graph is not symmetric about the origin (e.g. $(3, 2)$ is on the graph but $(-3, -2)$ is not)

\end{flushleft}

\vspace{2in}

\item $3x-2y=10$ \\ Re-write as: $y = \frac{3x-10}{2}$.

\begin{flushleft}

$x$-intercepts: $\left(\frac{10}{3}, 0 \right)$

$y$-intercept: $(0, -5)$

$\begin{array}{|r||c|c|}

\hline

x & y & (x,y) \\ \hline

-2 & -8 & (-2, -8) \\ \hline

-1 & -\frac{13}{2} & \left(-1, -\frac{13}{2}\right) \\ \hline

0 & -5 & (0, -5) \\ \hline

1 & -\frac{7}{2} & \left(1, -\frac{7}{2} \right) \\ \hline

2 & -2 & (2, -2) \\ \hline

\end{array} $

\begin{mfpic}[10]{-4}{5}{-10}{3}

\point[3pt]{(-2,-8), (-1,-6.5), (0,-5), (1,-3.5), (2,-2), (3.333,0)}

\axes

\tlabel[cc](5,-0.5){\scriptsize $x$}

\tlabel[cc](0.5,3){\scriptsize $y$}

\xmarks{-3,-2,-1,1,2,3,4}

\ymarks{-9,-8,-7,-6,-5,-4,-3,-2,-1,1,2}

\tlpointsep{4pt}

\axislabels {x}{{\tiny $-3 \hspace{6pt}$} -3,{\tiny $-2 \hspace{6pt}$} -2, {\tiny $-1 \hspace{6pt}$} -1, {\tiny $1$} 1, {\tiny $2$} 2, {\tiny $3$} 3, {\tiny $4$} 4}

\axislabels {y}{{\tiny $-9$} -9,{\tiny $-8$} -8,{\tiny $-7$} -7,{\tiny $-6$} -6,{\tiny $-5$} -5,{\tiny $-4$} -4,{\tiny $-3$} -3,{\tiny $-2$} -2,{\tiny $-1$} -1, {\tiny $1$} 1, {\tiny $2$} 2}

\arrow \reverse \arrow \function{-3, 4.5, 0.1}{(3*x-10)/2}

\end{mfpic}

The graph is not symmetric about the $x$-axis (e.g. $(2, -2)$ is on the graph but $(2,2)$ is not)

The graph is not symmetric about the $y$-axis (e.g. $(2, -2)$ is on the graph but $(-2, -2)$ is not)

The graph is not symmetric about the origin (e.g. $(2, -2)$ is on the graph but $(-2, 2)$ is not)

\end{flushleft}

\pagebreak

\item $(x+2)^2+y^2=16$ \\ Re-write as $y = \pm \sqrt{16-(x+2)^2}$.

\begin{flushleft}

$x$-intercepts: $(-6, 0)$, $(2,0)$

$y$-intercepts: $\left(0, \pm 2\sqrt{3}\right)$

$\begin{array}{|r||c|c|}

\hline

x & y & (x,y) \\ \hline

-6 & 0 & (-6,0) \\ \hline

-4 & \pm 2 \sqrt{3} & \left(-4,\pm 2 \sqrt{3}\right) \\ \hline

-2 & \pm 4 & (-2, \pm 4) \\ \hline

0 & \pm 2 \sqrt{3} & \left(0,\pm 2 \sqrt{3}\right) \\ \hline

2 & 0 & (2, 0) \\ \hline

\end{array} $

\begin{mfpic}[10]{-8}{4}{-6}{6}

\point[3pt]{(-6,0), (-4, 3.4641), (-4, -3.4641), (-2,4), (-2,-4), (0, 3.4641), (0, -3.4641), (2,0) }

\axes

\tlabel[cc](4,-0.5){\scriptsize $x$}

\tlabel[cc](0.5,6){\scriptsize $y$}

\xmarks{-7,-6,-5,-4,-3,-2,-1,1,2,3}

\ymarks{-5,-4,-3,-2,-1,1,2,3,4,5}

\tlpointsep{4pt}

\axislabels {x}{{\tiny $-7 \hspace{6pt}$} -7,{\tiny $-6 \hspace{6pt}$} -6, {\tiny $-5 \hspace{6pt}$} -5,{\tiny $-4 \hspace{6pt}$} -4, {\tiny $-3 \hspace{6pt}$} -3,{\tiny $-2 \hspace{6pt}$} -2, {\tiny $-1 \hspace{6pt}$} -1, {\tiny $1$} 1, {\tiny $2$} 2, {\tiny $3$} 3}

\axislabels {y}{{\tiny $-5$} -5, {\tiny $-4$} -4, {\tiny $-3$} -3, {\tiny $-2$} -2, {\tiny $-1$} -1, {\tiny $1$} 1, {\tiny $2$} 2, {\tiny $3$} 3, {\tiny $4$} 4, {\tiny $5$} 5}

\circle{(-2,0),4}

\end{mfpic}

The graph is symmetric about the $x$-axis

The graph is not symmetric about the $y$-axis (e.g. $(-6, 0)$ is on the graph but $(6, 0)$ is not)

The graph is not symmetric about the origin (e.g. $(-6, 0)$ is on the graph but $(6, 0)$ is not)

\end{flushleft}

\vspace{2in}

\item $x^{2} - y^{2} = 1$ \\ Re-write as: $y = \pm \sqrt{x^{2} - 1}$.

\begin{flushleft}

$x$-intercepts: $(-1, 0), (1, 0)$

The graph has no $y$-intercepts

$\begin{array}{|r||c|c|}

\hline

x & y & (x,y) \\ \hline

-3 & \pm \sqrt{8} & (-3, \pm \sqrt{8}) \\ \hline

-2 & \pm \sqrt{3} & (-2, \pm \sqrt{3}) \\ \hline

-1 & 0 & (-1, 0) \\ \hline

1 & 0 & (1, 0) \\ \hline

2 & \pm \sqrt{3} & (2, \pm \sqrt{3}) \\ \hline

3 & \pm \sqrt{8} & (3, \pm \sqrt{8}) \\ \hline

\end{array} $

\begin{mfpic}[10]{-4}{4}{-4}{4}

\point[3pt]{(-3,2.828), (-3,-2.828),(-2,1.732),(-2,-1.732),(-1,0),(1, 0),(3,2.828),(3,-2.828),(2,1.732),(2, -1.732)}

\axes

\tlabel[cc](4,-0.5){\scriptsize $x$}

\tlabel[cc](0.5,4){\scriptsize $y$}

\xmarks{-3,-2,-1,1,2,3}

\ymarks{-3,-2,-1,1,2,3}

\tlpointsep{4pt}

\axislabels {x}{{\tiny $-3 \hspace{6pt}$} -3, {\tiny $-2 \hspace{6pt}$} -2, {\tiny $-1 \hspace{6pt}$} -1, {\tiny $1$} 1, {\tiny $2$} 2, {\tiny $3$} 3}

\axislabels {y}{{\tiny $-3$} -3, {\tiny $-2$} -2, {\tiny $-1$} -1, {\tiny $1$} 1, {\tiny $2$} 2, {\tiny $3$} 3}

\arrow \reverse \arrow \parafcn{-2,2,0.1}{(cosh(t),sinh(t))}

\arrow \reverse \arrow \parafcn{-2,2,0.1}{(-cosh(t),sinh(t))}

\end{mfpic}

The graph is symmetric about the $x$-axis

The graph is symmetric about the $y$-axis

The graph is symmetric about the origin

\end{flushleft}

\pagebreak

\item $4y^2-9x^2 = 36$ \\

Re-write as: $y = \pm \frac{\sqrt{9x^2+36}}{2}$.

\begin{flushleft}

The graph has no $x$-intercepts

$y$-intercepts: $(0, \pm 3)$

$\begin{array}{|r||c|c|}

\hline

x & y & (x,y) \\ \hline

-4 & \pm 3 \sqrt{5} & \left(-4,\pm 3 \sqrt{5}\right) \\ \hline

-2 & \pm 3 \sqrt{2} & \left(-2,\pm 3 \sqrt{2}\right) \\ \hline

0 & \pm 3 & (0, \pm 3) \\ \hline

2 & \pm 3 \sqrt{2} & \left(2,\pm 3 \sqrt{2}\right) \\ \hline

4 & \pm 3 \sqrt{5} & \left(4,\pm 3 \sqrt{5}\right) \\ \hline

\end{array}$

\begin{mfpic}[10]{-5}{5}{-8}{8}

\point[3pt]{(-4, 6.708), (4, 6.708), (-2, 4.243), (2, 4.243), (0,3), (0,-3),(-4, -6.708), (4, -6.708), (-2, -4.243), (2, -4.243) }

\axes

\tlabel[cc](5,-0.5){\scriptsize $x$}

\tlabel[cc](0.5,8){\scriptsize $y$}

\xmarks{-4,-3,-2,-1, 1, 2, 3, 4}

\ymarks{-7,-6,-5,-4,-3,-2,-1,1,2,3,4,5,6,7}

\tlpointsep{4pt}

\axislabels {x}{{\tiny $-4 \hspace{6pt}$} -4,{\tiny $-3 \hspace{6pt}$} -3,{\tiny $-2 \hspace{6pt}$} -2, {\tiny $-1 \hspace{6pt}$} -1, {\tiny $1$} 1, {\tiny $2$} 2, {\tiny $3$} 3, {\tiny $4$} 4}

\axislabels {y}{{\tiny $-7$} -7, {\tiny $-6$} -6,{\tiny $-5$} -5,{\tiny $-4$} -4,{\tiny $-3$} -3,{\tiny $-2$} -2,{\tiny $-1$} -1,{\tiny $1$} 1,{\tiny $2$} 2,{\tiny $3$} 3,{\tiny $4$} 4,{\tiny $5$} 5,{\tiny $6$} 6,{\tiny $7$} 7 }

\arrow \reverse \arrow \parafcn{-1.6,1.6,0.1}{(2*sinh(t), 3*cosh(t))}

\arrow \reverse \arrow \parafcn{-1.6,1.6,0.1}{(2*sinh(t), 0-3*cosh(t))}

\end{mfpic}

The graph is symmetric about the $x$-axis

The graph is symmetric about the $y$-axis

The graph is symmetric about the origin

\end{flushleft}

\vspace{2in}

\item $x^{3}y = -4$ \\ Re-write as: $y = -\dfrac{4}{x^{3}}$.

\begin{flushleft}

The graph has no $x$-intercepts

The graph has no $y$-intercepts

$\begin{array}{|r||c|c|}

\hline

x & y & (x,y) \\ \hline

-2 & \frac{1}{2} & (-2, \frac{1}{2}) \\ \hline

-1 & 4 & (-1, 4) \\ \hline

-\frac{1}{2} & 32 & (-\frac{1}{2}, 32) \\ \hline

\frac{1}{2} & -32 & (\frac{1}{2}, -32)\\ \hline

1 & -4 & (1, -4) \\ \hline

2 & -\frac{1}{2} & (2, -\frac{1}{2}) \\ \hline

\end{array} $

\begin{mfpic}[10]{-5}{5}{-9}{9}

\point[3pt]{(-4,0.125), (-2,1), (-1, 8), (1, -8), (2, -1), (4, -0.125)}

\axes

\tlabel[cc](5,-0.5){\scriptsize $x$}

\tlabel[cc](0.5,9){\scriptsize $y$}

\xmarks{-4,-2,2,4}

\ymarks{-8,-1,1,8}

\tlpointsep{4pt}

\axislabels {x}{{\tiny $-2 \hspace{6pt}$} -4, {\tiny $-1 \hspace{6pt}$} -2, {\tiny $1$} 2, {\tiny $2$} 4}

\axislabels {y}{{\tiny $-32$} -8, {\tiny $-4$} -1, {\tiny $4$} 1, {\tiny $32$} 8}

\arrow \reverse \arrow \function{-4.5, -0.95, 0.1}{-8/(x**3)}

\arrow \reverse \arrow \function{0.95, 4.5, 0.1}{-8/(x**3)}

\end{mfpic}

The graph is not symmetric about the $x$-axis (e.g. $(1, -4)$ is on the graph but $(1, 4)$ is not)

The graph is not symmetric about the $y$-axis (e.g. $(1, -4)$ is on the graph but $(-1, -4)$ is not)

The graph is symmetric about the origin

\end{flushleft}

\end{enumerate}

\end{multicols}

\end{enumerate}

\closegraphsfile

1.7: Transformations

Suppose \((2,-3)\) is on the graph of \(y = f(x)\). In Exercises \ref{transformpointfirst} - \ref{transformpointlast}, use Theorem \ref{transformationsthm} to find a point on the graph of the given transformed function.

\begin{multicols}{3}

\begin{enumerate}

\item \(y = f(x)+3\) \label{transformpointfirst}

\item \(y = f(x+3)$$

\item \(y = f(x)-1$$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item \(y = f(x-1)$$

\item \(y = 3f(x)$$

\item \(y = f(3x)$$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item \(y = -f(x)$$

\item \(y = f(-x)$$

\item \(y = f(x-3)+1$$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item \(y = 2f(x+1)$$

\item \(y = 10 - f(x)$$

\item \(y = 3f(2x) - 1$$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item \(y = \frac{1}{2} f(4-x)$$

\item \(y = 5f(2x+1) + 3$$

\item \(y = 2f(1-x) -1$$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item \(y =f\left(\dfrac{7-2x}{4}\right)$$

\item \(y = \dfrac{f(3x) - 1}{2}$$

\item \(y = \dfrac{4-f(3x-1)}{7}\) \label{transformpointlast}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

The complete graph of \(y = f(x)\) is given below. In Exercises \ref{transformgraphfirst} - \ref{transformgraphlast}, use it and Theorem \ref{transformationsthm} to graph the given transformed function.

\vspace{-.1in}

\begin{center}

\begin{mfpic}[15]{-5}{5}{-1}{5}

\axes

\arrow \reverse \arrow \polyline{(-4,4), (0,0), (4,4)}

\point[3pt]{(-2,2), (0,0), (2,2)}

\tlabel[cc](5,-0.25){\scriptsize \(x$$}

\tlabel[cc](0.25,5){\scriptsize \(y$$}

\tlabel[cc](-2.5,1.25){\scriptsize \((-2,2)$$}

\tlabel[cc](0.75,-0.5){\scriptsize \((0,0)$$}

\tlabel[cc](2.25,1.25){\scriptsize \((2,2)$$}

\tcaption{The graph for Ex. \ref{transformgraphfirst} - \ref{transformgraphlast}}

\xmarks{-4,-3,-2,-1,2,3,4}

\ymarks{1,2,3,4}

\tlpointsep{5pt}

\scriptsize

\axislabels {x}{{\)-4 \hspace{7pt}$$} -4,{\)-3 \hspace{7pt}$$} -3, {\)-2 \hspace{7pt}$$} -2, {\)-1 \hspace{7pt}$$} -1, {$$2$$} 2,{$$3$$} 3,{$$4$$} 4}

\axislabels {y}{{$$1$$} 1, {$$2$$} 2, {$$3$$} 3, {$$4$$} 4}

\normalsize

\end{mfpic}

\end{center}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item \(y = f(x) + 1\) \label{transformgraphfirst}

\item \(y = f(x) - 2$$

\item \(y = f(x+1)$$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item \(y = f(x - 2)$$

\item \(y = 2f(x)$$

\item \(y = f(2x)$$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item \(y = 2 - f(x)$$

\item \(y = f(2-x)$$

\item \(y = 2-f(2-x)\) \label{transformgraphlast}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item Some of the answers to Exercises \ref{transformgraphfirst} - \ref{transformgraphlast} above should be the same. Which ones match up? What properties of the graph of \(y=f(x)\) contribute to the duplication?

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\pagebreak

The complete graph of \(y = f(x)\) is given below. In Exercises \ref{transsecondgraphfirst} - \ref{transsecondgraphlast}, use it and Theorem \ref{transformationsthm} to graph the given transformed function.

\vspace{-.1in}

\begin{center}

\begin{mfpic}[15]{-5}{5}{-5}{5}

\axes

\polyline{(-2,0), (0,4), (2,0), (4,-2)}

\point[3pt]{(-2,0), (0,4), (2,0), (4,-2)}

\tlabel[cc](5,-0.25){\scriptsize \(x$$}

\tlabel[cc](0.25,5){\scriptsize \(y$$}

\tlabel[cc](-2.25,-1.25){\scriptsize \((-2,0)$$}

\tlabel[cc](1,4){\scriptsize \((0,4)$$}

\tlabel[cc](2,-1.25){\scriptsize \((2,0)$$}

\tlabel[cc](4,-2.5){\scriptsize \((4,-2)$$}

\tcaption{The graph for Ex. \ref{transsecondgraphfirst} - \ref{transsecondgraphlast}}

\xmarks{-4,-3,-2,-1,1,2,3,4}

\ymarks{-4,-3,-2,-1,1,2,3,4}

\tlpointsep{5pt}

\scriptsize

\axislabels {x}{{\)-4 \hspace{7pt}$$} -4,{\)-3 \hspace{7pt}$$} -3, {\)-1 \hspace{7pt}$$} -1,{$$1$$} 1,{$$3$$} 3,{$$4$$} 4}

\axislabels {y}{{\)-4$$} -4,{\)-3$$} -3,{\)-2$$} -2, {\)-1$$} -1, {$$1$$} 1, {$$2$$} 2, {$$3$$} 3, {$$4$$} 4}

\normalsize

\end{mfpic}

\end{center}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item \(y = f(x) - 1\) \label{transsecondgraphfirst}

\item \(y = f(x + 1)$$

\item \(y = \frac{1}{2} f(x)$$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item \(y = f(2x)$$

\item \(y = - f(x)$$

\item \(y = f(-x)$$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item \(y = f(x+1) - 1$$

\item \(y = 1 - f(x)$$

\item \(y = \frac{1}{2}f(x+1)-1\) \label{transsecondgraphlast}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

The complete graph of \(y = f(x)\) is given below. In Exercises \ref{transthirdgraphfirst} - \ref{transthirdgraphlast}, use it and Theorem \ref{transformationsthm} to graph the given transformed function.

\vspace{-.1in}

\begin{center}

\begin{mfpic}[20]{-4}{4}{-1.5}{4}

\point[3pt]{(-3,0),(3,0),(0,3)}

\parafcn{0,3.14159,0.1}{(3*cos(t), 3*sin(t))}

\tlabel[cc](-3,-1){\small \(\left(-3, 0 \right)$$}

\tlabel[cc](0.8,3.3){\small \(\left(0, 3 \right)$$}

\tlabel[cc](3,-1){\small \(\left(3, 0 \right)$$}

\axes

\tlabel[cc](4,-0.5){\scriptsize \(x$$}

\tlabel[cc](0.5,4){\scriptsize \(y$$}

\tcaption{The graph for Ex. \ref{transthirdgraphfirst} - \ref{transthirdgraphlast}}

\xmarks{-3,-2,-1,1,2,3}

\ymarks{-1,1,2,3}

\tlpointsep{5pt}

\scriptsize

\axislabels {x}{{\)-3 \hspace{7pt}$$} -3, {\)-2 \hspace{7pt}$$} -2, {\)-1 \hspace{7pt}$$} -1, {$$1$$} 1, {$$2$$} 2, {$$3$$} 3}

\axislabels {y}{{\)-1$$} -1, {$$1$$} 1, {$$2$$} 2, {$$3$$} 3}

\normalsize

\end{mfpic}

\end{center}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item \(g(x) = f(x) + 3\) \label{transthirdgraphfirst}

\item \(h(x) = f(x) - \frac{1}{2}$$

\item \(j(x) = f\left(x - \frac{2}{3}\right)$$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item \(a(x) = f(x + 4)$$

\item \(b(x) = f(x + 1) - 1$$

\item \(c(x) = \frac{3}{5}f(x)$$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item \(d(x) = -2f(x)$$

\item \(k(x) = f\left(\frac{2}{3}x\right)$$

\item \(m(x) = -\frac{1}{4}f(3x)$$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item \(n(x) = 4f(x - 3) - 6$$

\item \(p(x) = 4 + f(1 - 2x)$$

\item \(q(x) = -\frac{1}{2}f\left(\frac{x + 4}{2}\right) - 3\) \label{transthirdgraphlast}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\pagebreak

The complete graph of \(y = S(x)\) is given below.

\vspace{-.1in}

\begin{center}

\begin{mfpic}[20]{-3}{3}{-4}{4}

\axes

\function{-2,2,0.1}{3*sin(1.570796327*x)}

\point[3pt]{(-2,0), (-1,-3), (0,0), (1,3), (2,0)}

\tlabel[cc](3,-0.25){\scriptsize \(x$$}

\tlabel[cc](0.25,4){\scriptsize \(y$$}

\tlabel[cc](-2,0.5){\scriptsize \((-2,0)$$}

\tlabel[cc](-1,-3.5){\scriptsize \((-1,-3)$$}

\tlabel[cc](0.5,0.25){\scriptsize \((0,0)$$}

\tlabel[cc](1,3.5){\scriptsize \((1,3)$$}

\tlabel[cc](2,-0.5){\scriptsize \((2,0)$$}

\tcaption{The graph of \(y=S(x)$$}

\xmarks{-2,-1,1,2}

\ymarks{-3,-2,-1,1,2,3}

\tlpointsep{5pt}

\scriptsize

\axislabels {x}{{\)-2 \hspace{7pt}$$} -2,{\)-1 \hspace{7pt}$$} -1,{$$1$$} 1}

\axislabels {y}{{\)-3$$} -3,{\)-2$$} -2, {\)-1$$} -1, {$$1$$} 1, {$$2$$} 2, {$$3$$} 3}

\normalsize

\end{mfpic}

\end{center}

The purpose of Exercises \ref{transformsinegraphfirst} - \ref{transformsinegraphlast} is to graph \(y = \frac{1}{2}S(-x+1) + 1\) by graphing each transformation, one step at a time.

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item \(y = S_{\text{\tiny \(1$$}}(x) = S(x + 1)\) \label{transformsinegraphfirst}

\item \(y = S_{\text{\tiny \(2$$}}(x) = S_{\text{\tiny \(1$$}}(-x) = S(-x + 1)$$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item \(y = S_{\text{\tiny \(3$$}}(x) = \frac{1}{2} S_{\text{\tiny \(2$$}}(x) = \frac{1}{2}S(-x+1)$$

\item \(y = S_{\text{\tiny \(4$$}}(x) = S_{\text{\tiny \(3$$}}(x) + 1 = \frac{1}{2}S(-x+1) + 1\) \label{transformsinegraphlast}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

Let \(f(x) = \sqrt{x}\). Find a formula for a function \(g\) whose graph is obtained from \(f\) from the given sequence of transformations.

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item (1) shift right 2 units; (2) shift down 3 units

\item (1) shift down 3 units; (2) shift right 2 units

\item (1) reflect across the \(x\)-axis; (2) shift up 1 unit

\item (1) shift up 1 unit; (2) reflect across the \(x\)-axis

\item (1) shift left 1 unit; (2) reflect across the \(y\)-axis; (3) shift up 2 units

\item (1) reflect across the \(y\)-axis; (2) shift left 1 unit; (3) shift up 2 units

\item (1) shift left 3 units; (2) vertical stretch by a factor of 2; (3) shift down 4 units

\item (1) shift left 3 units; (2) shift down 4 units; (3) vertical stretch by a factor of 2

\item (1) shift right 3 units; (2) horizontal shrink by a factor of 2; (3) shift up 1 unit

\item (1) horizontal shrink by a factor of 2; (2) shift right 3 units; (3) shift up 1 unit

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item The graph of \(y = f(x) = \sqrt[3]{x}\) is given below on the left and the graph of \(y = g(x)\) is given on the right. Find a formula for \(g\) based on transformations of the graph of \(f\). Check your answer by confirming that the points shown on the graph of \(g\) satisfy the equation \(y = g(x)\).

\[ \begin{array}{cc}

\begin{mfpic}[10]{-12}{9}{-6}{6}

\point[3pt]{(0,0), (-1, -1), (1, 1), (-8, -2), (8, 2)}

\axes

\tlabel[cc](9,-0.5){\scriptsize \(x$$}

\tlabel[cc](0.5,6){\scriptsize \(y$$}

\xmarks{-11 step 1 until 8}

\ymarks{-5 step 1 until 5}

\tlpointsep{4pt}

\axislabels {x}{{\tiny \(-11 \hspace{6pt}$$} -11, {\tiny \(-10 \hspace{6pt}$$} -10, {\tiny \(-9 \hspace{6pt}$$} -9, {\tiny \(-8 \hspace{6pt}$$} -8, {\tiny \(-7 \hspace{6pt}$$} -7, {\tiny \(-6 \hspace{6pt}$$} -6, {\tiny \(-5 \hspace{6pt}$$} -5, {\tiny \(-4 \hspace{6pt}$$} -4, {\tiny \(-3 \hspace{6pt}$$} -3, {\tiny \(-2 \hspace{6pt}$$} -2, {\tiny \(-1 \hspace{6pt}$$} -1, {\tiny \(1$$} 1, {\tiny \(2$$} 2, {\tiny \(3$$} 3, {\tiny \(4$$} 4, {\tiny \(5$$} 5, {\tiny \(6$$} 6, {\tiny \(7$$} 7, {\tiny \(8$$} 8}

\axislabels {y}{{\tiny \(-5$$} -5, {\tiny \(-4$$} -4, {\tiny \(-3$$} -3, {\tiny \(-2$$} -2, {\tiny \(-1$$} -1, {\tiny \(1$$} 1, {\tiny \(2$$} 2, {\tiny \(3$$} 3, {\tiny \(4$$} 4, {\tiny \(5$$} 5}

\arrow \reverse \arrow \parafcn{-2.1,2.1,0.1}{(t**3,t)}

\tcaption{\scriptsize \(y = \sqrt[3]{x}$$}

\end{mfpic}

&

\begin{mfpic}[10]{-12}{9}{-6}{6}

\point[3pt]{(-11,3), (-4,1), (-3,-1), (-2,-3), (5,-5)}

\axes

\tlabel[cc](9,-0.5){\scriptsize \(x$$}

\tlabel[cc](0.5,6){\scriptsize \(y$$}

\xmarks{-11 step 1 until 8}

\ymarks{-5 step 1 until 5}

\tlpointsep{4pt}

\axislabels {x}{{\tiny \(-11 \hspace{6pt}$$} -11, {\tiny \(-10 \hspace{6pt}$$} -10, {\tiny \(-9 \hspace{6pt}$$} -9, {\tiny \(-8 \hspace{6pt}$$} -8, {\tiny \(-7 \hspace{6pt}$$} -7, {\tiny \(-6 \hspace{6pt}$$} -6, {\tiny \(-5 \hspace{6pt}$$} -5, {\tiny \(-4 \hspace{6pt}$$} -4, {\tiny \(-3 \hspace{6pt}$$} -3, {\tiny \(-2 \hspace{6pt}$$} -2, {\tiny \(-1 \hspace{6pt}$$} -1, {\tiny \(1$$} 1, {\tiny \(2$$} 2, {\tiny \(3$$} 3, {\tiny \(4$$} 4, {\tiny \(5$$} 5, {\tiny \(6$$} 6, {\tiny \(7$$} 7, {\tiny \(8$$} 8}

\axislabels {y}{{\tiny \(-5$$} -5, {\tiny \(-4$$} -4, {\tiny \(-3$$} -3, {\tiny \(-2$$} -2, {\tiny \(-1$$} -1, {\tiny \(1$$} 1, {\tiny \(2$$} 2, {\tiny \(3$$} 3, {\tiny \(4$$} 4, {\tiny \(5$$} 5}

\arrow \reverse \arrow \parafcn{-2.1,2.1,0.1}{((t**3 - 3),((-2*t) - 1))}

\tcaption{\scriptsize \(y = g(x)$$}

\end{mfpic}

\end{array} \]

\item For many common functions, the properties of Algebra make a horizontal scaling the same as a vertical scaling by (possibly) a different factor. For example, we stated earlier that \(\sqrt{9x} = 3\sqrt{x}\). With the help of your classmates, find the equivalent vertical scaling produced by the horizontal scalings \(y = (2x)^{3}, \, y = |5x|, \, y = \sqrt[3]{27x} \, \( and \(\, y = \left(\frac{1}{2} x\right)^{2}\). What about \(y = (-2x)^{3}, \, y = |-5x|, \, y = \sqrt[3]{-27x}\, \( and \(\, y = \left(-\frac{1}{2} x\right)^{2}\)?

\item We mentioned earlier in the section that, in general, the order in which transformations are applied matters, yet in our first example with two transformations the order did not matter. (You could perform the shift to the left followed by the shift down or you could shift down and then left to achieve the same result.) With the help of your classmates, determine the situations in which order does matter and those in which it does not.

\item What happens if you reflect an even function across the \(y\)-axis?

\item What happens if you reflect an odd function across the \(y\)-axis?

\item What happens if you reflect an even function across the \(x\)-axis?

\item What happens if you reflect an odd function across the \(x\)-axis?

\item How would you describe symmetry about the origin in terms of reflections?

\item As we saw in Example \ref{graphingcalctrans}, the viewing window on the graphing calculator affects how we see the transformations done to a graph. Using two different calculators, find viewing windows so that \(f(x) = x^{2}\) on the one calculator looks like \(g(x) = 3x^{2}\) on the other.

\end{enumerate}

\newpage

\subsection{Answers}

\begin{multicols}{3}

\begin{enumerate}

\item \((2,0)$$

\item \((-1,-3)$$

\item \((2,-4)$$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item \((3,-3)$$

\item \((2,-9)$$

\item \(\left(\frac{2}{3}, -3\right)$$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item \((2,3)$$

\item \((-2,-3)$$

\item \((5,-2)$$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item \((1,-6)$$

\item \((2,13)$$

\item \(y = (1,-10)$$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item \(\left(2, -\frac{3}{2}\right)$$

\item \(\left(\frac{1}{2}, -12 \right)$$

\item \((-1,-7)$$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item \(\left(-\frac{1}{2}, -3\right)$$

\item \(\left(\frac{2}{3}, -2 \right)$$

\item \((1,1)$$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item \(y = f(x) + 1$$

\begin{mfpic}[15]{-5}{5}{-1}{5}

\axes

\arrow \reverse \arrow \polyline{(-4,5), (0,1), (4,5)}

\point[3pt]{(-2,3), (0,1), (2,3)}

\tlabel[cc](5,-0.25){\scriptsize \(x$$}

\tlabel[cc](0.25,5){\scriptsize \(y$$}

\tlabel[cc](-2.5,2.25){\scriptsize \((-2,3)$$}

\tlabel[cc](0.75,0.5){\scriptsize \((0,1)$$}

\tlabel[cc](2.25,2.25){\scriptsize \((2,3)$$}

\xmarks{-4,-3,-2,-1,1,2,3,4}

\ymarks{1,2,3,4}

\tlpointsep{5pt}

\scriptsize

\axislabels {x}{{\)-4 \hspace{7pt}$$} -4,{\)-3 \hspace{7pt}$$} -3, {\)-2 \hspace{7pt}$$} -2, {\)-1 \hspace{7pt}$$} -1, {$$1$$} 1, {$$2$$} 2,{$$3$$} 3,{$$4$$} 4}

\axislabels {y}{{$$1$$} 1, {$$2$$} 2, {$$3$$} 3, {$$4$$} 4}

\normalsize

\end{mfpic}

\vfill

\columnbreak

\item \(y = f(x) - 2$$

\begin{mfpic}[15]{-5}{5}{-3}{3}

\axes

\arrow \reverse \arrow \polyline{(-4,2), (0,-2), (4,2)}

\point[3pt]{(-2,0), (0,-2), (2,0)}

\tlabel[cc](5,-0.25){\scriptsize \(x$$}

\tlabel[cc](0.25,3){\scriptsize \(y$$}

\tlabel[cc](-2.5,-0.75){\scriptsize \((-2,2)$$}

\tlabel[cc](0.75,-2.5){\scriptsize \((0,-2)$$}

\tlabel[cc](2.25,-0.75){\scriptsize \((2,2)$$}

\xmarks{-4,-3,-2,-1,1,2,3,4}

\ymarks{-2,-1,1,2}

\tlpointsep{5pt}

\scriptsize

\axislabels {x}{{\)-4 \hspace{7pt}$$} -4, {\)-1 \hspace{7pt}$$} -1, {$$1$$} 1,{$$4$$} 4}

\axislabels {y}{{\)-2$$} -2, {\)-1$$} -1, {$$1$$} 1, {$$2$$} 2}

\normalsize

\end{mfpic}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item \(y = f(x+1)$$

\begin{mfpic}[15]{-6}{4}{-1}{5}

\axes

\arrow \reverse \arrow \polyline{(-5,4), (-1,0), (3,4)}

\point[3pt]{(-3,2), (-1,0), (1,2)}

\tlabel[cc](4,-0.25){\scriptsize \(x$$}

\tlabel[cc](0.25,5){\scriptsize \(y$$}

\tlabel[cc](-3.5,1.25){\scriptsize \((-3,2)$$}

\tlabel[cc](-1,-0.5){\scriptsize \((-1,0)$$}

\tlabel[cc](1.25,1.25){\scriptsize \((1,2)$$}

\xmarks{-5,-4,-3,-2,-1,1,2,3}

\ymarks{1,2,3,4}

\tlpointsep{5pt}

\scriptsize

\axislabels {x}{{\)-5 \hspace{7pt}$$} -5,{\)-4 \hspace{7pt}$$} -4,{\)-3 \hspace{7pt}$$} -3, {$$1$$} 1, {$$2$$} 2,{$$3$$} 3}

\axislabels {y}{{$$1$$} 1, {$$2$$} 2, {$$3$$} 3, {$$4$$} 4}

\normalsize

\end{mfpic}

\vfill

\columnbreak

\item \(y = f(x - 2)$$

\begin{mfpic}[15]{-3}{7}{-1}{5}

\axes

\arrow \reverse \arrow \polyline{(-2,4), (2,0), (6,4)}

\point[3pt]{(0,2), (2,0), (4,2)}

\tlabel[cc](7,-0.25){\scriptsize \(x$$}

\tlabel[cc](0.25,5){\scriptsize \(y$$}

\tlabel[cc](0.75,2){\scriptsize \((0,2)$$}

\tlabel[cc](2,-0.5){\scriptsize \((2,0)$$}

\tlabel[cc](3,2){\scriptsize \((4,2)$$}

\xmarks{-2,-1,1,2,3,4,5,6}

\ymarks{1,2,3,4}

\tlpointsep{5pt}

\scriptsize

\axislabels {x}{{\)-2 \hspace{7pt}$$} -2, {\)-1 \hspace{7pt}$$} -1, {$$3$$} 3,{$$4$$} 4,{$$5$$} 5,{$$6$$} 6}

\axislabels {y}{{$$1$$} 1, {$$2$$} 2, {$$4$$} 4}

\normalsize

\end{mfpic}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item \(y = 2f(x)$$

\begin{mfpic}[15]{-5}{5}{-1}{5}

\axes

\arrow \reverse \arrow \polyline{(-2.5,5), (0,0), (2.5,5)}

\point[3pt]{(-2,4), (0,0), (2,4)}

\tlabel[cc](5,-0.25){\scriptsize \(x$$}

\tlabel[cc](0.25,5){\scriptsize \(y$$}

\tlabel[cc](-2.75,3.5){\scriptsize \((-2,4)$$}

\tlabel[cc](0.75,-0.5){\scriptsize \((0,0)$$}

\tlabel[cc](2.5,3.5){\scriptsize \((2,4)$$}

\xmarks{-4,-3,-2,-1,2,3,4}

\ymarks{1,2,3,4}

\tlpointsep{5pt}

\scriptsize

\axislabels {x}{{\)-4 \hspace{7pt}$$} -4,{\)-3 \hspace{7pt}$$} -3, {\)-2 \hspace{7pt}$$} -2, {\)-1 \hspace{7pt}$$} -1, {$$2$$} 2,{$$3$$} 3,{$$4$$} 4}

\axislabels {y}{{$$1$$} 1, {$$2$$} 2, {$$3$$} 3, {$$4$$} 4}

\normalsize

\end{mfpic}

\vfill

\columnbreak

\item \(y = f(2x)$$

\begin{mfpic}[15]{-5}{5}{-1}{5}

\axes

\arrow \reverse \arrow \polyline{(-2.5,5), (0,0), (2.5,5)}

\point[3pt]{(-1,2), (0,0), (1,2)}

\tlabel[cc](5,-0.25){\scriptsize \(x$$}

\tlabel[cc](0.25,5){\scriptsize \(y$$}

\tlabel[cc](-1.75,1.5){\scriptsize \((-1,2)$$}

\tlabel[cc](0.75,-0.5){\scriptsize \((0,0)$$}

\tlabel[cc](1.5,1.5){\scriptsize \((1,2)$$}

\xmarks{-4,-3,-2,-1,2,3,4}

\ymarks{1,2,3,4}

\tlpointsep{5pt}

\scriptsize

\axislabels {x}{{\)-4 \hspace{7pt}$$} -4,{\)-3 \hspace{7pt}$$} -3, {\)-2 \hspace{7pt}$$} -2, {\)-1 \hspace{7pt}$$} -1, {$$2$$} 2,{$$3$$} 3,{$$4$$} 4}

\axislabels {y}{{$$1$$} 1, {$$2$$} 2, {$$3$$} 3, {$$4$$} 4}

\normalsize

\end{mfpic}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item \(y = 2 - f(x)$$

\begin{mfpic}[15]{-5}{5}{-2}{3}

\axes

\arrow \reverse \arrow \polyline{(-4,-2), (0,2), (4,-2)}

\point[3pt]{(-2,0), (0,2), (2,0)}

\tlabel[cc](5,-0.25){\scriptsize \(x$$}

\tlabel[cc](0.25,3){\scriptsize \(y$$}

\tlabel[cc](-1.25,-0.5){\scriptsize \((-2,0)$$}

\tlabel[cc](0.75,2){\scriptsize \((0,2)$$}

\tlabel[cc](1.5,-0.5){\scriptsize \((2,0)$$}

\xmarks{-4,-3,-2,-1,2,3,4}

\ymarks{-1,1,2}

\tlpointsep{5pt}

\scriptsize

\axislabels {x}{{\)-4 \hspace{7pt}$$} -4,{\)-3 \hspace{7pt}$$} -3, {$$3$$} 3,{$$4$$} 4}

\axislabels {y}{{$$1$$} 1, {$$2$$} 2}

\normalsize

\end{mfpic}

\vfill

\columnbreak

\item \(y = f(2-x)$$

\begin{mfpic}[15]{-3}{7}{-1}{5}

\axes

\arrow \reverse \arrow \polyline{(-2,4), (2,0), (6,4)}

\point[3pt]{(0,2), (2,0), (4,2)}

\tlabel[cc](7,-0.25){\scriptsize \(x$$}

\tlabel[cc](0.25,5){\scriptsize \(y$$}

\tlabel[cc](0.75,2){\scriptsize \((0,2)$$}

\tlabel[cc](2,-0.5){\scriptsize \((2,0)$$}

\tlabel[cc](3,2){\scriptsize \((4,2)$$}

\xmarks{-2,-1,1,2,3,4,5,6}

\ymarks{1,2,3,4}

\tlpointsep{5pt}

\scriptsize

\axislabels {x}{{\)-2 \hspace{7pt}$$} -2, {\)-1 \hspace{7pt}$$} -1, {$$3$$} 3,{$$4$$} 4,{$$5$$} 5,{$$6$$} 6}

\axislabels {y}{{$$1$$} 1, {$$2$$} 2, {$$4$$} 4}

\normalsize

\end{mfpic}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item \(y = 2-f(2-x)$$

\begin{mfpic}[15]{-3}{7}{-1}{5}

\axes

\arrow \reverse \arrow \polyline{(-2,-2), (2,2), (6,-2)}

\point[3pt]{(0,0), (2,2), (4,0)}

\tlabel[cc](7,-0.25){\scriptsize \(x$$}

\tlabel[cc](0.25,5){\scriptsize \(y$$}

\tlabel[cc](0.75,-0.5){\scriptsize \((0,0)$$}

\tlabel[cc](2,2.5){\scriptsize \((2,2)$$}

\tlabel[cc](3.5,-0.5){\scriptsize \((4,0)$$}

\xmarks{-2,-1,1,2,3,4,5,6}

\ymarks{1,2,3,4}

\tlpointsep{5pt}

\scriptsize

\axislabels {x}{{\)-2 \hspace{7pt}$$} -2, {\)-1 \hspace{7pt}$$} -1, {$$2$$} 2,{$$5$$} 5, {$$6$$} 6}

\axislabels {y}{{$$1$$} 1, {$$2$$} 2, {$$3$$} 3, {$$4$$} 4}

\normalsize

\end{mfpic}

\vfill

\columnbreak

\addtocounter{enumi}{1}

\item \(y = f(x) - 1$$

\begin{mfpic}[15]{-5}{5}{-5}{5}

\axes

\polyline{(-2,-1), (0,3), (2,-1), (4,-3)}

\point[3pt]{(-2,-1), (0,3), (2,-1), (4,-3)}

\tlabel[cc](5,-0.25){\scriptsize \(x$$}

\tlabel[cc](0.25,5){\scriptsize \(y$$}

\tlabel[cc](-2.25,-1.5){\scriptsize \((-2,-1)$$}

\tlabel[cc](1,3){\scriptsize \((0,3)$$}

\tlabel[cc](1,-1.5){\scriptsize \((2,-1)$$}

\tlabel[cc](4,-3.5){\scriptsize \((4,-3)$$}

\xmarks{-4,-3,-2,-1,1,2,3,4}

\ymarks{-4,-3,-2,-1,1,2,3,4}

\tlpointsep{5pt}

\scriptsize

\axislabels {x}{{\)-4 \hspace{7pt}$$} -4,{\)-3 \hspace{7pt}$$} -3, {\)-1 \hspace{7pt}$$} -1,{\)-2 \hspace{7pt}$$} -2,{$$1$$} 1,{$$2$$} 2,{$$3$$} 3,{$$4$$} 4}

\axislabels {y}{{\)-4$$} -4,{\)-3$$} -3,{\)-2$$} -2, {\)-1$$} -1, {$$1$$} 1, {$$2$$} 2, {$$3$$} 3, {$$4$$} 4}

\normalsize

\end{mfpic}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item \(y = f(x + 1)$$

\begin{mfpic}[15]{-5}{5}{-5}{5}

\axes

\polyline{(-3,0), (-1,4), (1,0), (3,-2)}

\point[3pt]{(-3,0), (-1,4), (1,0), (3,-2)}

\tlabel[cc](5,-0.25){\scriptsize \(x$$}

\tlabel[cc](0.25,5){\scriptsize \(y$$}

\tlabel[cc](-3.25,-1.25){\scriptsize \((-3,0)$$}

\tlabel[cc](-3,4){\scriptsize \((-1,4)$$}

\tlabel[cc](1,-1.25){\scriptsize \((1,0)$$}

\tlabel[cc](3,-2.5){\scriptsize \((3,-2)$$}

\xmarks{-4,-3,-2,-1,1,2,3,4}

\ymarks{-4,-3,-2,-1,1,2,3,4}

\tlpointsep{5pt}

\scriptsize

\axislabels {x}{{\)-4 \hspace{7pt}$$} -4,{\)-3 \hspace{7pt}$$} -3, {\)-1 \hspace{7pt}$$} -1,{\)-2 \hspace{7pt}$$} -2,{$$1$$} 1,{$$2$$} 2,{$$3$$} 3,{$$4$$} 4}

\axislabels {y}{{\)-4$$} -4,{\)-3$$} -3,{\)-2$$} -2, {\)-1$$} -1, {$$1$$} 1, {$$2$$} 2, {$$3$$} 3, {$$4$$} 4}

\normalsize

\end{mfpic}

\vfill

\columnbreak

\item \(y = \frac{1}{2} f(x)$$

\begin{mfpic}[15]{-5}{5}{-5}{5}

\axes

\polyline{(-2,0), (0,2), (2,0), (4,-1)}

\point[3pt]{(-2,0), (0,2), (2,0), (4,-1)}

\tlabel[cc](5,-0.25){\scriptsize \(x$$}

\tlabel[cc](0.25,5){\scriptsize \(y$$}

\tlabel[cc](-2.25,-1.25){\scriptsize \((-2,0)$$}

\tlabel[cc](1,2){\scriptsize \((0,2)$$}

\tlabel[cc](2,-1.25){\scriptsize \((2,0)$$}

\tlabel[cc](4,-1.5){\scriptsize \((4,-1)$$}

\xmarks{-4,-3,-2,-1,1,2,3,4}

\ymarks{-4,-3,-2,-1,1,2,3,4}

\tlpointsep{5pt}

\scriptsize

\axislabels {x}{{\)-4 \hspace{7pt}$$} -4,{\)-3 \hspace{7pt}$$} -3, {\)-1 \hspace{7pt}$$} -1,{$$1$$} 1,{$$3$$} 3,{$$4$$} 4}

\axislabels {y}{{\)-4$$} -4,{\)-3$$} -3,{\)-2$$} -2, {\)-1$$} -1, {$$1$$} 1, {$$2$$} 2, {$$3$$} 3, {$$4$$} 4}

\normalsize

\end{mfpic}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\pagebreak

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item \(y = f(2x)$$

\begin{mfpic}[15]{-5}{5}{-5}{5}

\axes

\polyline{(-1,0), (0,4), (1,0), (2,-2)}

\point[3pt]{(-1,0), (0,4), (1,0), (2,-2)}

\tlabel[cc](5,-0.25){\scriptsize \(x$$}

\tlabel[cc](0.25,5){\scriptsize \(y$$}

\tlabel[cc](-1,-0.75){\scriptsize \((-1,0)$$}

\tlabel[cc](1,4){\scriptsize \((0,4)$$}

\tlabel[cc](1.75,0.5){\scriptsize \((1,0)$$}

\tlabel[cc](2,-2.5){\scriptsize \((2,-2)$$}

\xmarks{-4,-3,-2,-1,1,2,3,4}

\ymarks{-4,-3,-2,-1,1,2,3,4}

\tlpointsep{5pt}

\scriptsize

\axislabels {x}{{\)-4 \hspace{7pt}$$} -4,{\)-3 \hspace{7pt}$$} -3, {\)-2 \hspace{7pt}$$} -2,{$$2$$} 2,{$$3$$} 3,{$$4$$} 4}

\axislabels {y}{{\)-4$$} -4,{\)-3$$} -3,{\)-2$$} -2, {$$1$$} 1, {$$2$$} 2, {$$3$$} 3, {$$4$$} 4}

\normalsize

\end{mfpic}

\vfill

\columnbreak

\item \(y = - f(x)$$

\begin{mfpic}[15]{-5}{5}{-5}{5}

\axes

\polyline{(-2,0), (0,-4), (2,0), (4,2)}

\point[3pt]{(-2,0), (0,-4), (2,0), (4,2)}

\tlabel[cc](5,-0.25){\scriptsize \(x$$}

\tlabel[cc](0.25,5){\scriptsize \(y$$}

\tlabel[cc](-2.25,.75){\scriptsize \((-2,0)$$}

\tlabel[cc](1.25,-4){\scriptsize \((0,-4)$$}

\tlabel[cc](1.75,.75){\scriptsize \((2,0)$$}

\tlabel[cc](4,2.5){\scriptsize \((4,2)$$}

\xmarks{-4,-3,-2,-1,1,2,3,4}

\ymarks{-4,-3,-2,-1,1,2,3,4}

\tlpointsep{5pt}

\scriptsize

\axislabels {x}{{\)-4 \hspace{7pt}$$} -4,{\)-3 \hspace{7pt}$$} -3, {\)-1 \hspace{7pt}$$} -1,{\)-2 \hspace{7pt}$$} -2,{$$1$$} 1,{$$2$$} 2,{$$3$$} 3,{$$4$$} 4}

\axislabels {y}{{\)-4$$} -4,{\)-3$$} -3,{\)-2$$} -2, {\)-1$$} -1, {$$1$$} 1, {$$2$$} 2, {$$3$$} 3, {$$4$$} 4}

\normalsize

\end{mfpic}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item \(y = f(-x)$$

\begin{mfpic}[15]{-5}{5}{-5}{5}

\axes

\polyline{(2,0), (0,4), (-2,0), (-4,-2)}

\point[3pt]{(2,0), (0,4), (-2,0), (-4,-2)}

\tlabel[cc](5,-0.25){\scriptsize \(x$$}

\tlabel[cc](0.25,5){\scriptsize \(y$$}

\tlabel[cc](2.25,-1.25){\scriptsize \((2,0)$$}

\tlabel[cc](1,4){\scriptsize \((0,4)$$}

\tlabel[cc](-2,-1.25){\scriptsize \((-2,0)$$}

\tlabel[cc](-4,-2.5){\scriptsize \((-4,-2)$$}

\xmarks{-4,-3,-2,-1,1,2,3,4}

\ymarks{-4,-3,-2,-1,1,2,3,4}

\tlpointsep{5pt}

\scriptsize

\axislabels {x}{{\)-4 \hspace{7pt}$$} -4,{\)-3 \hspace{7pt}$$} -3, {\)-1 \hspace{7pt}$$} -1,{$$1$$} 1,{$$3$$} 3,{$$4$$} 4}

\axislabels {y}{{\)-4$$} -4,{\)-3$$} -3,{\)-2$$} -2, {\)-1$$} -1, {$$1$$} 1, {$$2$$} 2, {$$3$$} 3, {$$4$$} 4}

\normalsize

\end{mfpic}

\vfill

\columnbreak

\item \(y = f(x+1) - 1$$

\begin{mfpic}[15]{-5}{5}{-5}{5}

\axes

\polyline{(-3,-1), (-1,3), (1,-1), (3,-3)}

\point[3pt]{(-3,-1), (-1,3), (1,-1), (3,-3)}

\tlabel[cc](5,-0.25){\scriptsize \(x$$}

\tlabel[cc](0.25,5){\scriptsize \(y$$}

\tlabel[cc](-3.25,-2.25){\scriptsize \((-3,-1)$$}

\tlabel[cc](-3,3){\scriptsize \((-1,3)$$}

\tlabel[cc](2.5,-1){\scriptsize \((1,-1)$$}

\tlabel[cc](3,-3.5){\scriptsize \((3,-3)$$}

\xmarks{-4,-3,-2,-1,1,2,3,4}

\ymarks{-4,-3,-2,-1,1,2,3,4}

\tlpointsep{5pt}

\scriptsize

\axislabels {x}{{\)-4 \hspace{7pt}$$} -4,{\)-3 \hspace{7pt}$$} -3, {\)-1 \hspace{7pt}$$} -1,{\)-2 \hspace{7pt}$$} -2,{$$1$$} 1,{$$2$$} 2,{$$3$$} 3,{$$4$$} 4}

\axislabels {y}{{\)-4$$} -4,{\)-3$$} -3,{\)-2$$} -2, {\)-1$$} -1, {$$1$$} 1, {$$2$$} 2, {$$3$$} 3, {$$4$$} 4}

\normalsize

\end{mfpic}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item \(y = 1 - f(x)$$

\begin{mfpic}[15]{-5}{5}{-5}{5}

\axes

\polyline{(-2,1), (0,-3), (2,1), (4,3)}

\point[3pt]{(-2,1), (0,-3), (2,1), (4,3)}

\tlabel[cc](5,-0.25){\scriptsize \(x$$}

\tlabel[cc](0.25,5){\scriptsize \(y$$}

\tlabel[cc](-2.25,1.75){\scriptsize \((-2,1)$$}

\tlabel[cc](1.25,-3){\scriptsize \((0,-3)$$}

\tlabel[cc](1.75,1.75){\scriptsize \((2,1)$$}

\tlabel[cc](4,3.5){\scriptsize \((4,3)$$}

\xmarks{-4,-3,-2,-1,1,2,3,4}

\ymarks{-4,-3,-2,-1,1,2,3,4}

\tlpointsep{5pt}

\scriptsize

\axislabels {x}{{\)-4 \hspace{7pt}$$} -4,{\)-3 \hspace{7pt}$$} -3, {\)-1 \hspace{7pt}$$} -1,{\)-2 \hspace{7pt}$$} -2,{$$1$$} 1,{$$2$$} 2,{$$3$$} 3,{$$4$$} 4}

\axislabels {y}{{\)-4$$} -4,{\)-3$$} -3,{\)-2$$} -2, {\)-1$$} -1, {$$1$$} 1, {$$2$$} 2, {$$3$$} 3, {$$4$$} 4}

\normalsize

\end{mfpic}

\vfill

\columnbreak

\item \(y = \frac{1}{2}f(x+1)-1$$

\begin{mfpic}[15]{-5}{5}{-5}{5}

\axes

\polyline{(-3,-1), (-1,1), (1,-1), (3,-2)}

\point[3pt]{(-3,-1), (-1,1), (1,-1), (3,-2)}

\tlabel[cc](5,-0.25){\scriptsize \(x$$}

\tlabel[cc](0.25,5){\scriptsize \(y$$}

\tlabel[cc](-3.25,-1.5){\scriptsize \((-3,-1)$$}

\tlabel[cc](-2.25,1){\scriptsize \((-1,1)$$}

\tlabel[cc](2.5,-1){\scriptsize \((1,-1)$$}

\tlabel[cc](3,-2.5){\scriptsize \((3,-2)$$}

\xmarks{-4,-3,-2,-1,1,2,3,4}

\ymarks{-4,-3,-2,-1,1,2,3,4}

\tlpointsep{5pt}

\scriptsize

\axislabels {x}{{\)-4 \hspace{7pt}$$} -4,{\)-3 \hspace{7pt}$$} -3, {\)-1 \hspace{7pt}$$} -1,{\)-2 \hspace{7pt}$$} -2,{$$1$$} 1,{$$2$$} 2,{$$3$$} 3,{$$4$$} 4}

\axislabels {y}{{\)-4$$} -4,{\)-3$$} -3,{\)-2$$} -2, {\)-1$$} -1, {$$1$$} 1, {$$2$$} 2, {$$3$$} 3, {$$4$$} 4}

\normalsize

\end{mfpic}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item \(g(x) = f(x) + 3$$\\

\begin{mfpic}[15]{-4}{4}{-1.5}{7}

\point[3pt]{(-3,3),(3,3),(0,6)}

\parafcn{0,3.14159,0.1}{(3*cos(t), (3*sin(t)) + 3)}

%\function{-3,3,.1}{3 + sqrt(9 - (x**2))}

\tlabel[cc](-3,2){\tiny \(\left(-3, 3 \right)$$}

\tlabel[cc](0.8,6.3){\tiny \(\left(0, 6 \right)$$}

\tlabel[cc](3,2){\tiny \(\left(3, 3 \right)$$}

\axes

\tlabel[cc](4,-0.5){\scriptsize \(x$$}

\tlabel[cc](0.5,7){\scriptsize \(y$$}

\xmarks{-3,-2,-1,1,2,3}

\ymarks{-1,1,2,3,4,5,6}

\tlpointsep{4pt}

\tiny

\axislabels {x}{{\)-3 \hspace{7pt}$$} -3, {\)-2 \hspace{7pt}$$} -2, {\)-1 \hspace{7pt}$$} -1, {$$1$$} 1, {$$2$$} 2, {$$3$$} 3}

\axislabels {y}{{\)-1$$} -1, {$$1$$} 1, {$$2$$} 2, {$$3$$} 3, {$$4$$} 4, {$$5$$} 5, {$$6$$} 6}

\normalsize

\end{mfpic}

\vfill

\columnbreak

\item \(h(x) = f(x) - \frac{1}{2}$$\\

\begin{mfpic}[15]{-4}{4}{-1.5}{4}

\point[3pt]{(-3,-0.5),(3,-0.5),(0,2.5)}

%\function{-3,3,.1}{sqrt(9 - (x**2)) - 0.5}

\parafcn{0,3.14159,0.1}{(3*cos(t), (3*sin(t)) - 0.5)}

\tlabel[cc](-3,-1){\tiny \(\left(-3, -\frac{1}{2} \right)$$}

\tlabel[cc](0.8,3){\tiny \(\left(0, \frac{5}{2} \right)$$}

\tlabel[cc](3,-1){\tiny \(\left(3, -\frac{1}{2} \right)$$}

\axes

\tlabel[cc](4,-0.5){\scriptsize \(x$$}

\tlabel[cc](0.5,4){\scriptsize \(y$$}

\xmarks{-3,-2,-1,1,2,3}

\ymarks{-1,1,2,3}

\tlpointsep{4pt}

\tiny

\axislabels {x}{{\)-3 \hspace{7pt}$$} -3, {\)-2 \hspace{7pt}$$} -2, {\)-1 \hspace{7pt}$$} -1, {$$1$$} 1, {$$2$$} 2, {$$3$$} 3}

\axislabels {y}{{\)-1$$} -1, {$$1$$} 1, {$$2$$} 2, {$$3$$} 3}

\normalsize

\end{mfpic}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item \(j(x) = f\left(x - \frac{2}{3}\right)$$\\

\begin{mfpic}[15]{-4}{4.5}{-1.5}{4}

\point[3pt]{(-2.333,0),(3.6667,0),(.6667, 3)}

%\function{-2.333,3.6667,.1}{sqrt(9 - ((x - 0.6667)**2))}

\parafcn{0,3.14159,0.1}{((3*cos(t)) + 0.6667, 3*sin(t))}

\tlabel[cc](-2.333,-1){\tiny \(\left(-\frac{7}{3}, 0 \right)$$}

\tlabel[cc](1.5,3.5){\tiny \(\left(\frac{2}{3}, 3 \right)$$}

\tlabel[cc](3.6667,-1){\tiny \(\left(\frac{11}{3}, 0 \right)$$}

\axes

\tlabel[cc](4.5,-0.5){\scriptsize \(x$$}

\tlabel[cc](0.5,4){\scriptsize \(y$$}

\xmarks{-3,-2,-1,1,2,3}

\ymarks{-1,1,2,3}

\tlpointsep{4pt}

\tiny

\axislabels {x}{{\)-3 \hspace{7pt}$$} -3, {\)-2 \hspace{7pt}$$} -2, {\)-1 \hspace{7pt}$$} -1, {$$1$$} 1, {$$2$$} 2, {$$3$$} 3}

\axislabels {y}{{\)-1$$} -1, {$$1$$} 1, {$$2$$} 2, {$$3$$} 3}

\normalsize

\end{mfpic}

\vfill

\columnbreak

\item \(a(x) = f(x + 4)$$\\

\begin{mfpic}[15]{-8}{1}{-1.5}{4}

\point[3pt]{(-7,0),(-1,0),(-4, 3)}

%\function{-7,-1,.1}{sqrt(9 - ((x + 4)**2))}

\parafcn{0,3.14159,0.1}{((3*cos(t)) - 4, 3*sin(t))}

\tlabel[cc](-7,-1){\tiny \(\left(-7, 0 \right)$$}

\tlabel[cc](-3,3.5){\tiny \(\left(-4, 3 \right)$$}

\tlabel[cc](-1,-1){\tiny \(\left(-1, 0 \right)$$}

\axes

\tlabel[cc](1,-0.5){\scriptsize \(x$$}

\tlabel[cc](0.5,4){\scriptsize \(y$$}

\xmarks{-7,-6,-5,-4,-3,-2,-1}

\ymarks{-1,1,2,3}

\tlpointsep{4pt}

\tiny

\axislabels {x}{{\)-7 \hspace{7pt}$$} -7, {\)-6 \hspace{7pt}$$} -6, {\)-5 \hspace{7pt}$$} -5, {\)-4 \hspace{7pt}$$} -4, {\)-3 \hspace{7pt}$$} -3, {\)-2 \hspace{7pt}$$} -2, {\)-1 \hspace{7pt}$$} -1}

\axislabels {y}{{$$1$$} 1, {$$2$$} 2, {$$3$$} 3}

\normalsize

\end{mfpic}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item \(b(x) = f(x + 1) - 1$$\\

\begin{mfpic}[15]{-5}{3}{-2}{3}

\point[3pt]{(-4,-1),(-1,2),(2,-1)}

%\function{-4,2,.1}{sqrt(9 - ((x + 1)**2)) - 1}

\parafcn{0,3.14159,0.1}{((3*cos(t)) - 1, (3*sin(t)) - 1)}

\tlabel[cc](-4,-1.5){\tiny \(\left(-4, -1 \right)$$}

\tlabel[cc](-1.5,2.5){\tiny \(\left(-1,2 \right)$$}

\tlabel[cc](2,-1.5){\tiny \(\left(2, -1 \right)$$}

\axes

\tlabel[cc](3,-0.5){\scriptsize \(x$$}

\tlabel[cc](0.5,3){\scriptsize \(y$$}

\xmarks{-4,-3,-2,-1,1,2}

\ymarks{-1,1,2}

\tlpointsep{4pt}

\tiny

\axislabels {x}{{\)-4 \hspace{7pt}$$} -4, {\)-3 \hspace{7pt}$$} -3, {\)-2 \hspace{7pt}$$} -2, {\)-1 \hspace{7pt}$$} -1, {$$1$$} 1, {$$2$$} 2}

\axislabels {y}{{\)-1$$} -1, {$$1$$} 1, {$$2$$} 2}

\normalsize

\end{mfpic}

\vfill

\columnbreak

\item \(c(x) = \frac{3}{5}f(x)$$\\

\begin{mfpic}[15]{-4}{4}{-1.5}{3}

\point[3pt]{(-3,0),(3,0),(0,1.8)}

%\function{-3,3,.1}{0.6*sqrt(9 - (x**2))}

\parafcn{0,3.14159,0.1}{(3*cos(t), 1.8*sin(t))}

\tlabel[cc](-3,-1){\tiny \(\left(-3, 0 \right)$$}

\tlabel[cc](0.8,2.3){\tiny \(\left(0, \frac{9}{5} \right)$$}

\tlabel[cc](3,-1){\tiny \(\left(3, 0 \right)$$}

\axes

\tlabel[cc](4,-0.5){\scriptsize \(x$$}

\tlabel[cc](0.5,3){\scriptsize \(y$$}

\xmarks{-3,-2,-1,1,2,3}

\ymarks{-1,1,2}

\tlpointsep{4pt}

\tiny

\axislabels {x}{{\)-3 \hspace{7pt}$$} -3, {\)-2 \hspace{7pt}$$} -2, {\)-1 \hspace{7pt}$$} -1, {$$1$$} 1, {$$2$$} 2, {$$3$$} 3}

\axislabels {y}{{\)-1$$} -1, {$$1$$} 1, {$$2$$} 2}

\normalsize

\end{mfpic}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item \(d(x) = -2f(x)$$\\

\begin{mfpic}[15]{-4}{4}{-7}{1}

\point[3pt]{(-3,0),(3,0),(0,-6)}

%\function{-3,3,.1}{-2*sqrt(9 - (x**2))}

\parafcn{0,3.14159,0.1}{(3*cos(t), -6*sin(t))}

\tlabel[cc](-3,0.5){\tiny \(\left(-3, 0 \right)$$}

\tlabel[cc](0.8,-6.5){\tiny \(\left(0, -6 \right)$$}

\tlabel[cc](3,0.5){\tiny \(\left(3, 0 \right)$$}

\axes

\tlabel[cc](4,-0.5){\scriptsize \(x$$}

\tlabel[cc](0.5,1){\scriptsize \(y$$}

\xmarks{-3,-2,-1,1,2,3}

\ymarks{-6,-5,-4,-3,-2,-1}

\tlpointsep{4pt}

\tiny

\axislabels {x}{{\)-3 \hspace{7pt}$$} -3, {\)-2 \hspace{7pt}$$} -2, {\)-1 \hspace{7pt}$$} -1, {$$1$$} 1, {$$2$$} 2, {$$3$$} 3}

\axislabels {y}{{\)-6$$} -6, {\)-5$$} -5, {\)-4$$} -4, {\)-3$$} -3, {\)-2$$} -2, {\)-1$$} -1}

\normalsize

\end{mfpic}

\vfill

\columnbreak

\item \(k(x) = f\left(\frac{2}{3}x\right)$$\\

\begin{mfpic}[15]{-5}{5}{-1.5}{4}

\point[3pt]{(-4.5,0),(4.5,0),(0,3)}

%\function{-4.5,4.5,.1}{sqrt(9 - ((0.66666*x)**2))}

\parafcn{0,3.14159,0.1}{(4.5*cos(t), 3*sin(t))}

\tlabel[cc](-4.5,-1){\tiny \(\left(-\frac{9}{2}, 0 \right)$$}

\tlabel[cc](0.8,3.5){\tiny \(\left(0, 3 \right)$$}

\tlabel[cc](4.5,-1){\tiny \(\left(\frac{9}{2}, 0 \right)$$}

\axes

\tlabel[cc](5,-0.5){\scriptsize \(x$$}

\tlabel[cc](0.5,4){\scriptsize \(y$$}

\xmarks{-4,-3,-2,-1,1,2,3,4}

\ymarks{-1,1,2,3}

\tlpointsep{4pt}

\tiny

\axislabels {x}{{\)-4 \hspace{7pt}$$} -4, {\)-3 \hspace{7pt}$$} -3, {\)-2 \hspace{7pt}$$} -2, {\)-1 \hspace{7pt}$$} -1, {$$1$$} 1, {$$2$$} 2, {$$3$$} 3, {$$4$$} 4}

\axislabels {y}{{\)-1$$} -1, {$$1$$} 1, {$$2$$} 2, {$$3$$} 3}

\normalsize

\end{mfpic}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\pagebreak

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item \(m(x) = -\frac{1}{4}f(3x)$$\\

\begin{mfpic}[30]{-2}{2}{-1.5}{1}

\point[3pt]{(-1,0),(1,0),(0,-0.75)}

%\function{-1,1,.1}{-0.25*sqrt(9 - ((3*x)**2))}

\parafcn{0,3.14159,0.1}{(cos(t), -0.75*sin(t))}

\tlabel[cc](-1,0.25){\scriptsize \(\left( -1, 0 \right)$$}

\tlabel[cc](0.8,-1){\scriptsize \(\left(0, -\frac{3}{4} \right)$$}

\tlabel[cc](1,0.25){\scriptsize \(\left( 1, 0 \right)$$}

\axes

\tlabel[cc](2,-0.25){\scriptsize \(x$$}

\tlabel[cc](0.25,1){\scriptsize \(y$$}

\xmarks{-1,1}

\ymarks{-1}

\tlpointsep{4pt}

\tiny

\axislabels {x}{{\)-1 \hspace{7pt}$$} -1, {$$1$$} 1}

\axislabels {y}{{\)-1$$} -1}

\normalsize

\end{mfpic}

\vfill

\columnbreak

\item \(n(x) = 4f(x - 3) - 6$$\\

\begin{mfpic}[15]{-1}{7}{-7}{7}

\point[3pt]{(0,-6),(3,6),(6,-6)}

%\function{0,6,.1}{4*sqrt(9 - ((x - 3)**2)) - 6}

\parafcn{0,3.14159,0.1}{(3*cos(t) + 3, (12*sin(t)) - 6)}

\tlabel[cc](1,-6.5){\tiny \(\left(0, -6 \right)$$}

\tlabel[cc](3,6.5){\tiny \(\left(3, 6 \right)$$}

\tlabel[cc](5.5,-6.5){\tiny \(\left(6, -6 \right)$$}

\axes

\tlabel[cc](7,-0.5){\scriptsize \(x$$}

\tlabel[cc](0.5,7){\scriptsize \(y$$}

\xmarks{1,2,3,4,5,6}

\ymarks{-6 step 1 until 6}

\tlpointsep{4pt}

\tiny

\axislabels {x}{{$$1$$} 1, {$$2$$} 2, {$$3$$} 3, {$$4$$} 4, {$$5$$} 5, {$$6$$} 6}

\axislabels {y}{{\)-6$$} -6, {\)-5$$} -5, {\)-4$$} -4, {\)-3$$} -3, {\)-2$$} -2, {\)-1$$} -1, {$$1$$} 1, {$$2$$} 2, {$$3$$} 3, {$$4$$} 4, {$$5$$} 5, {$$6$$} 6}

\normalsize

\end{mfpic}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item \(p(x) = 4 + f(1 - 2x) = f(-2x + 1) + 4$$\\

\begin{mfpic}[15]{-2}{3}{-1.5}{8}

\point[3pt]{(-1,4),(0.5,7),(2,4)}

%\function{-1,2,.1}{4 + sqrt(9 - (((-2*x) + 1)**2))}

\parafcn{0,3.14159,0.1}{((1 - 3*cos(t))/2, 3*sin(t) + 4)}

\tlabel[cc](-1,3.5){\tiny \(\left(-1, 4 \right)$$}

\tlabel[cc](1.5,7.2){\tiny \(\left(\frac{1}{2}, 7 \right)$$}

\tlabel[cc](2,3.5){\tiny \(\left(2, 4 \right)$$}

\axes

\tlabel[cc](3,-0.5){\scriptsize \(x$$}

\tlabel[cc](0.5,8){\scriptsize \(y$$}

\xmarks{-1,1,2}

\ymarks{-1,1,2,3,4,5,6,7}

\tlpointsep{4pt}

\tiny

\axislabels {x}{{\)-1 \hspace{7pt}$$} -1, {$$1$$} 1, {$$2$$} 2}

\axislabels {y}{{\)-1$$} -1, {$$1$$} 1, {$$2$$} 2, {$$3$$} 3, {$$4$$} 4, {$$5$$} 5, {$$6$$} 6, {$$7$$} 7}

\normalsize

\end{mfpic}

\vfill

\columnbreak

\item \small \(q(x) = -\frac{1}{2}f\left(\frac{x + 4}{2}\right) - 3 = -\frac{1}{2}f\left( \frac{1}{2}x + 2 \right) - 3 \(\\ \normalsize

\begin{mfpic}[10]{-11}{3}{-5.5}{1}

\point[3pt]{(-10,-3),(-4,-4.5),(2, -3)}

%\function{-10,2,.1}{-0.5*sqrt(9 - (((0.5*x) + 2)**2)) - 3}

\parafcn{0,3.14159,0.1}{(6*cos(t) - 4, -1.5*sin(t) - 3)}

\tlabel[cc](-10,-2.5){\tiny \(\left(-10, -3 \right)$$}

\tlabel[cc](-4,-5.25){\tiny \(\left(-4, -\frac{9}{2} \right)$$}

\tlabel[cc](2,-2.5){\tiny \(\left(2, -3 \right)$$}

\axes

\tlabel[cc](3,-0.5){\scriptsize \(x$$}

\tlabel[cc](0.5,1){\scriptsize \(y$$}

\xmarks{-10,-9,-8,-7,-6,-5,-4,-3,-2,-1,1,2}

\ymarks{-4,-3,-2,-1}

\tlpointsep{4pt}

\tiny

\axislabels {x}{{\)-10 \hspace{7pt}$$} -10, {\)-9 \hspace{7pt}$$} -9, {\)-8 \hspace{7pt}$$} -8, {\)-7 \hspace{7pt}$$} -7, {\)-6 \hspace{7pt}$$} -6, {\)-5 \hspace{7pt}$$} -5, {\)-4 \hspace{7pt}$$} -4, {\)-3 \hspace{7pt}$$} -3, {\)-2 \hspace{7pt}$$} -2, {\)-1 \hspace{7pt}$$} -1, {$$1$$} 1, {$$2$$} 2}

\axislabels {y}{{\)-4$$} -4, {\)-3$$} -3, {\)-2$$} -2, {\)-1$$} -1}

\normalsize

\end{mfpic}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\pagebreak

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item \(y = S_{\text{\tiny \(1$$}}(x) = S(x + 1)$$

\begin{mfpic}[20]{-4}{2}{-4}{4}

\axes

\function{-3,1,0.1}{3*sin(1.570796327*(x+1))}

\point[3pt]{(-3,0), (-2,-3), (-1,0), (0,3), (1,0)}

\tlabel[cc](2,-0.25){\scriptsize \(x$$}

\tlabel[cc](0.25,4){\scriptsize \(y$$}

\tlabel[cc](-3.5,0.5){\scriptsize \((-3,0)$$}

\tlabel[cc](-2,-3.5){\scriptsize \((-2,-3)$$}

\tlabel[cc](-1.75,0.5){\scriptsize \((-1,0)$$}

\tlabel[cc](0.75,3){\scriptsize \((0,3)$$}

\tlabel[cc](1,-0.5){\scriptsize \((1,0)$$}

\xmarks{-3,-2,-1,1}

\ymarks{-3,-2,-1,1,2,3}

\tlpointsep{5pt}

\scriptsize

\axislabels {x}{{\)-3 \hspace{7pt}$$} -3,{\)-2 \hspace{7pt}$$} -2,{\)-1 \hspace{7pt}$$} -1}

\axislabels {y}{{\)-3$$} -3,{\)-2$$} -2, {\)-1$$} -1, {$$1$$} 1, {$$2$$} 2, {$$3$$} 3}

\normalsize

\end{mfpic}

\vfill

\columnbreak

\item \(y = S_{\text{\tiny \(2$$}}(x) = S_{\text{\tiny \(1$$}}(-x) = S(-x + 1)$$

\begin{mfpic}[20]{-2}{4}{-4}{4}

\axes

\function{-1,3,0.1}{3*sin(1.570796327*(1-x))}

\point[3pt]{(3,0), (2,-3), (1,0), (0,3), (-1,0)}

\tlabel[cc](4,-0.25){\scriptsize \(x$$}

\tlabel[cc](0.25,4){\scriptsize \(y$$}

\tlabel[cc](3.5,0.5){\scriptsize \((3,0)$$}

\tlabel[cc](2,-3.5){\scriptsize \((2,-3)$$}

\tlabel[cc](1.75,0.5){\scriptsize \((1,0)$$}

\tlabel[cc](0.75,3){\scriptsize \((0,3)$$}

\tlabel[cc](-1,-0.5){\scriptsize \((-1,0)$$}

\xmarks{3,2,1,-1}

\ymarks{-3,-2,-1,1,2,3}

\tlpointsep{5pt}

\scriptsize

\axislabels {x}{ {$$1$$} 1, {$$2$$} 2, {$$3$$} 3}

\axislabels {y}{{\)-3$$} -3,{\)-2$$} -2, {\)-1$$} -1, {$$1$$} 1, {$$2$$} 2, {$$3$$} 3}

\normalsize

\end{mfpic}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item \(y = S_{\text{\tiny \(3$$}}(x) = \frac{1}{2} S_{\text{\tiny \(2$$}}(x) = \frac{1}{2}S(-x+1)$$

\begin{mfpic}[20]{-2}{4}{-3}{3}

\axes

\function{-1,3,0.1}{1.5*sin(1.570796327*(1-x))}

\point[3pt]{(3,0), (2,-1.5), (1,0), (0,1.5), (-1,0)}

\tlabel[cc](4,-0.25){\scriptsize \(x$$}

\tlabel[cc](0.25,3){\scriptsize \(y$$}

\tlabel[cc](3,0.5){\scriptsize \((3,0)$$}

\tlabel[cc](2,-2){\scriptsize \(\left(2,-\frac{3}{2} \right)$$}

\tlabel[cc](1.5,0.5){\scriptsize \((1,0)$$}

\tlabel[cc](0.75,1.5){\scriptsize \(\left(0,\frac{3}{2} \right)$$}

\tlabel[cc](-1,-0.5){\scriptsize \((-1,0)$$}

\xmarks{3,2,1,-1}

\ymarks{-2,-1,1,2}

\tlpointsep{5pt}

\scriptsize

\axislabels {x}{ {$$1$$} 1, {$$2$$} 2, {$$3$$} 3}

\axislabels {y}{{\)-2$$} -2,{\)-1$$} -1, {$$1$$} 1, {$$2$$} 2}

\normalsize

\end{mfpic}

\vfill

\columnbreak

\item \(y = S_{\text{\tiny \(4$$}}(x) = S_{\text{\tiny \(3$$}}(x) + 1 = \frac{1}{2}S(-x+1) + 1$$

\begin{mfpic}[20]{-2}{4}{-2}{4}

\axes

\function{-1,3,0.1}{1.5*sin(1.570796327*(1-x))+1}

\point[3pt]{(3,1), (2,-0.5), (1,1), (0,2.5), (-1,1)}

\tlabel[cc](4,-0.25){\scriptsize \(x$$}

\tlabel[cc](0.25,4){\scriptsize \(y$$}

\tlabel[cc](3,1.5){\scriptsize \((3,1)$$}

\tlabel[cc](2,-1){\scriptsize \(\left(2,-\frac{1}{2} \right)$$}

\tlabel[cc](1.5,1.5){\scriptsize \((1,1)$$}

\tlabel[cc](0.75,2.5){\scriptsize \(\left(0,\frac{5}{2} \right)$$}

\tlabel[cc](-1,.5){\scriptsize \((-1,1)$$}

\xmarks{3,2,1,-1}

\ymarks{-1,1,2,3}

\tlpointsep{5pt}

\scriptsize

\axislabels {x}{ {\)-1 \hspace{7pt}$$} -1,{$$1$$} 1, {$$3$$} 3}

\axislabels {y}{{\)-1$$} -1, {$$1$$} 1, {$$2$$} 2, {$$3$$} 3}

\normalsize

\end{mfpic}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item \(g(x) = \sqrt{x-2} - 3$$

\item \(g(x) = \sqrt{x-2} - 3$$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item \(g(x) = -\sqrt{x} + 1$$

\item \(g(x) = -(\sqrt{x} + 1) = -\sqrt{x} - 1$$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item \(g(x) = \sqrt{-x+1} + 2$$

\item \(g(x) = \sqrt{-(x+1)} + 2 = \sqrt{-x-1} + 2$$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item \(g(x) = 2\sqrt{x+3} - 4$$

\item \(g(x) = 2\left(\sqrt{x+3} - 4\right) = 2\sqrt{x+3} - 8$$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item \(g(x) = \sqrt{2x-3} + 1$$

\item \(g(x) = \sqrt{2(x-3)} + 1 = \sqrt{2x-6}+1$$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item \(g(x) = -2\sqrt[3]{x + 3} - 1\) or \(g(x) = 2\sqrt[3]{-x - 3} - 1$$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\closegraphsfile

Contributors

  • Carl Stitz, Ph.D. (Lakeland Community College) and Jeff Zeager, Ph.D. (Lorain County Community College)