27.1: Review of functions and graphs

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Page ID: 68482

Thomas Tradler and Holly Carley
CUNY New York City College of Technology via New York City College of Technology at CUNY Academic Works

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Exercise $\PageIndex{1}$

Find all solutions of the equation: $\big| 3x-9\big| =6$

Answer: $x = 1$ or $x = 5$

Exercise $\PageIndex{2}$

Find the equation of the line displayed below.

$clipboard_e6193cbb38730a86463aff4c9f1015643.png$

Answer: $y=-\dfrac{1}{2} x+1$

Exercise $\PageIndex{3}$

Find the solution of the equation: $x^3-3x^2+2x-2=0$ Approximate your answer to the nearest hundredth.

Answer: $x \approx 2.52$

Exercise $\PageIndex{4}$

Let $f(x)=x^2-2x+5$. Simplify the difference quotient $\dfrac{f(x+h)-f(x)}{h}$ as much as possible.

Answer: $2 x-2+2 h$

Exercise $\PageIndex{5}$

Consider the following graph of a function $f$.

$clipboard_ebbf8b3d341de54a74f708abe18207254.png$

Find: domain of $f$, range of $f$, $f(3)$, $f(5)$, $f(7)$, $f(9)$.

Answer: domain $D=[2,7]$, range $R=(1,4], f(3)=2, f(5)=2, f(7)=4, f(9) $ is undefined

Exercise $\PageIndex{6}$

Find the formula of the graph displayed below.

$clipboard_e79aaccb6dd8116b811f41f51239d7955.png$

Answer: $f(x)=-x^{2}+2$

Exercise $\PageIndex{7}$

Let $f(x)=5x+4$ and $g(x)=x^2+8x+7$. Find the quotient $\left(\dfrac f g\right)(x)$ and state its domain.

Answer: $\left(\dfrac{f}{g}\right)(x)=\dfrac{5 x+4}{x^{2}+8 x+7}=\dfrac{5 x+4}{(x+7)(x+1)}$ has domain $D=\mathbb{R}-\{-7,-1\}$

Exercise $\PageIndex{8}$

Let $f(x)=x^2+\sqrt{x-3}$ and $g(x)=2x-3$. Find the composition $(f\circ g)(x)$ and state its domain.

Answer: $(f \circ g)(x)=4 x^{2}-12 x+9+\sqrt{2 x-6}$ has domain $D=[3, \infty)$

Exercise $\PageIndex{9}$

Consider the assignments for $f$ and $g$ given by the table below.

\[\begin{array}{|c||c|c|c|c|c|}
\hline x & 2 & 3 & 4 & 5 & 6 \\
\hline f(x) & 5 & 0 & 2 & 4 & 2 \\
\hline g(x) & 6 & 2 & 3 & 4 & 1 \\
\hline
\end{array} \nonumber \]

Is $f$ a function? Is $g$ a function? Write the composed assignment for $(f\circ g)(x)$ as a table.

Answer

$f$ and $g$ are both functions, and the composition is given by the table:

$\begin{array}{|c||c|c|c|c|c|}
\hline x & 2 & 3 & 4 & 5 & 6 \\
\hline \hline f(x) & 5 & 0 & 2 & 4 & 2 \\
\hline g(x) & 6 & 2 & 3 & 4 & 1 \\
\hline(f \circ g)(x) & 2 & 5 & 0 & 2 & \text { undef. } \\
\hline
\end{array}$

Exercise $\PageIndex{10}$

Find the inverse of the function $f(x)=\dfrac{1}{2x+5}$.

Answer: $f^{-1}(x)=\dfrac{1}{2 x}-\dfrac{5}{2}$

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