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1.1: Sets of Real Numbers and the Cartesian Coordinate Plane
Exercise \(\PageIndex{1.1.1}\):
Fill in the chart below:
In Exercises 1.1.2  1.1.7, find the indicated intersection or union and simplify if possible. Express your answers in interval notation.
Exercise \(\PageIndex{1.1.3}\):
\( (1,1)\ \cup\ [0,6]\)
Exercise \(\PageIndex{1.1.4}\):
\( (\infty,4]\cap(0,\infty)\)
Exercise \(\PageIndex{1.1.5}\):
\( (\infty,0)\cap[1,5] \)
Exercise \(\PageIndex{1.1.6}\):
\( \( (\infty,0)\cup[1,5] \)\)
In Exercises 1.1.8  1.1.19, write the set using interval notation.
Exercise \(\PageIndex{1.1.9}\):
\( \{xx\neq1\}\)
Exercise \(\PageIndex{1.1.10}\):
\( \{xx\neq3,\ 4\}\)
Exercise \(\PageIndex{1.1.11}\):
\( \{xx\neq0,\ 2\}\)
Exercise \(\PageIndex{1.1.12}\):
\( \{x\,\, x \neq 2,\ 2 \}\)
Exercise \(\PageIndex{1.1.13}\):
\( \{xx\neq0,\ \pm4\}\)
Exercise \(\PageIndex{1.1.14}\):
\( \{xx\leq1\ \text{or}\ x\geq 1\}\)
Exercise \(\PageIndex{1.1.15}\):
\( \{xx<3\ \text{or}\ x\geq 2\}\)
Exercise \(\PageIndex{1.1.16}\):
\( \{xx\leq3\ \text{or}\ x>0\}\)
Exercise \(\PageIndex{1.1.17}\):
\( \{xx\leq5\ \text{or}\ x=6\}\)
Exercise \(\PageIndex{1.1.18}\):
\( \{xx>2\ \text{or}\ x=\pm1\}\)
Exercise \(\PageIndex{1.1.20}\):
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.
Exercise \(\PageIndex{1.1.21}\):
For each point given in Exercise 20 above:
 Identify the quadrant or axis in/on which the point lies.
 Find the point symmetric to the given point about the \(x\)axis.
 Find the point symmetric to the given point about the \(y\)axis.
 Find the point symmetric to the given point about the origin.
In Exercises 1.1.22  1.1.29, find the distance \(d\) between the points and the midpoint \(M\) of the line segment which connects them.
Exercise \(\PageIndex{1.1.22}\):
\( (1,\ 2),\ (3,\ 5)\)
Exercise \(\PageIndex{1.1.23}\):
\( (3,\ 10),\ (1,\ 2)\)
Exercise \(\PageIndex{1.1.24}\):
\( (\frac{1}{2},\ 4),\ (\frac{3}{2},\ 1)\)
Exercise \(\PageIndex{1.1.25}\):
\( (\frac{2}{3},\ \frac{3}{2}),\ (\frac{7}{3},\ 2)\)
Exercise \(\PageIndex{1.1.26}\):
\( (\frac{24}{5},\ \frac{6}{5}),\ (\frac{11}{5},\ \frac{19}{5})\)
Exercise \(\PageIndex{1.1.27}\):
\( (\sqrt{2},\ \sqrt{3}),\ (\sqrt{8},\sqrt{12})\)
Exercise \(\PageIndex{1.1.28}\):
\( (2\sqrt{45},\ \sqrt{12}),\ (\sqrt{20},\sqrt{27})\)
Exercise \(\PageIndex{1.1.29}\):
\((0,\ 0),\ (x,\ y) \)
Exercise \(\PageIndex{1.1.30}\):
Find all of the points of the form \((x,\ 1)\) which are 4 units from the point \((3,\ 2)\).
Exercise \(\PageIndex{1.1.31}\):
Find all of the points on the \( y\)axis which are 5 units from the point \((5,\ 3)\).
Exercise \(\PageIndex{1.1.32}\):
Find all of the points on the \( x\)axis which are 2 units from the point \((1,\ 1)\).
Exercise \(\PageIndex{1.1.33}\):
Find all of the points of the form \((x,\ x)\) which are 1 unit from the origin.
Exercise \(\PageIndex{1.1.34}\):
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 Sasquatchometer 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?
Exercise \(\PageIndex{1.1.35}\):
Add text here. For the automatic number to work, you need to add the "AutoNum" template (preferably at the end) to the page.
Exercise \(\PageIndex{A}\):
The points are arranged vertically. (Hint: Use \(P(a,\ y_0)\) and Q(a, y_1).)
Exercise \(\PageIndex{B}\):
The points are arranged horizontally. (Hint: Use \(P(x_0,\ b)\) and \(Q(x_1,\ b)\).)
Exercise \(\PageIndex{C}\):
The points are actually the same point. (You shouldn't need a hint for this one.)
Exercise \(\PageIndex{1.1.36}\):
Verify the Midpoint Formula by showing the distance between \(P(x_1,\ y_1)\) and \( M\) and the distance between \( M\) and \( Q(x_2,\ y2)\) are both half of the distance between \(P\) and \(Q\).
Exercise \(\PageIndex{1.1.37}\):
Show that the points \( A\), \( B\) and \( C\) below are the vertices of a right triangle.
Exercise \(\PageIndex{A}\):
\( A(3,2),\ B(6,4)\), and \(C(1,8)\)
Exercise \(\PageIndex{B}\):
\( A(3,\ 1),\ B(4,\ 0)\), and \(C(0,\ 3)\)
Exercise \(\PageIndex{1.1.38}\):
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.
Exercise \(\PageIndex{1.1.39}\):
Discuss with your classmates how many numbers are in the interval \( (0,\ 1)\).
Exercise \(\PageIndex{1.1.40}\):
The world is not at. 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?
1.2: Relations
In Exercises 1  20, graph the given relation.
Exercise \(\PageIndex{1}\):
\( \{(3, 9), (2, 4), (1, 1), (0, 0), (1, 1), (2, 4), (3, 9)\} \)
Exercise \(\PageIndex{2}\):
\( \{(2, 0), (1, 1), (1,1), (0, 2), (0,2), (1, 3), (1,3)\} \)
Exercise \(\PageIndex{3}\):
\(\{(m,\ 2m) m = 0;1,\pm2\}\)
Exercise \(\PageIndex{4}\):
\( \{(\frac{6}{k},\ k)\ k = \pm1,\pm2,\pm3,\pm4,\pm5,\pm6\}\)
Exercise \(\PageIndex{5}\):
\( \{(n,4n^2)\ n=0,\pm 1,\pm 2\} \)
Exercise \(\PageIndex{6}\):
\( \{(\sqrt{j},j)\ j=0,1,4,9\} \)
Exercise \(\PageIndex{7}\):
\( \{(x,2)\ x>4\} \)
Exercise \(\PageIndex{8}\):
\( \{(x,3)\ x\leq4\} \)
Exercise \(\PageIndex{9}\):
\( \{(1,y)\ y>1\} \)
Exercise \(\PageIndex{10}\):
\( \{(2,y)\ y\leq5\} \)
Exercise \(\PageIndex{11}\):
\( \{(2,y)\ 3<y\leq4\} \)
Exercise \(\PageIndex{12}\):
\( \{(3,y)\ 4\leq y<3\} \)
Exercise \(\PageIndex{13}\):
\( \{(x,2)\ 2\leq x<3\} \)
Exercise \(\PageIndex{14}\):
\( \{(x,3)\ 4<x\leq3\} \)
Exercise \(\PageIndex{15}\):
\( \{(x,y)\ x>2\} \)
Exercise \(\PageIndex{16}\):
\( \{(x,y)\ x\leq3\} \)
Exercise \(\PageIndex{17}\):
\( \{(x,y)\ y<4\} \)
Exercise \(\PageIndex{18}\):
\( \{(x,y)\ x\leq3,y<2\} \)
Exercise \(\PageIndex{19}\):
\( \{(x,y)\ x>0,y<4\} \)
Exercise \(\PageIndex{20}\):
\( \{(x,y)\ \sqrt{2}\leq x\leq \frac{2}{3},\pi <y\leq \frac{9}{2}\} \)
In Exercises 21  30, describe the given relation using either the roster or setbuilder method.
Exercise \(\PageIndex{21}\):
Exercise \(\PageIndex{22}\):
Exercise \(\PageIndex{23}\):
Exercise \(\PageIndex{24}\):
Exercise \(\PageIndex{25}\):
Exercise \(\PageIndex{26}\):
Exercise \(\PageIndex{27}\):
Exercise \(\PageIndex{28}\):
Exercise \(\PageIndex{29}\):
Exercise \(\PageIndex{30}\):
In Exercises 31  36, graph the given line.
Exercise \(\PageIndex{31}\):
\(x=2\)
Exercise \(\PageIndex{32}\):
\( x=3\)
Exercise \(\PageIndex{33}\):
\( y=3\)
Exercise \(\PageIndex{34}\):
\( y=2\)
Exercise \(\PageIndex{35}\):
\( x=0\)
Exercise \(\PageIndex{36}\):
\( y=0\)
Some relations are fairly easy to describe in words or with the roster method but are rather diffcult, if not impossible, to graph. Discuss with your classmates how you might graph the relations given in Exercises 37  40. Please note that in the notation below we are using the ellipsis, . . . , to denote that the list does not end, but rather, continues to follow the established pattern indenitely. For the relations in Exercises 37 and 38, give two examples of points which belong to the relation and two points which do not belong to the relation.
Exercise \(\PageIndex{37}\):
\( \{ (x,y)\ x\ \text{is an odd integer, and } y\ \text{is an even integer.}\}\)
Exercise \(\PageIndex{38}\):
\( \{(x,\ 1)\ x\text{ is an irrational number }\}\)
Exercise \(\PageIndex{39}\):
\( \{ (1,\ 0),\ (2,\ 1),\ (4,\ 2),\ (8,\ 3),\ (16,\ 4),\ (32,\ 5),...\} \)
Exercise \(\PageIndex{40}\):
\( \{...,(3,\ 9),(2,\ 4),(1,\ 1),(0,\ 0),(1,\ 1),(2,\ 4),(3,\ 9)...\} \)
For each equation given in Exercises 41  52:
 Find the x and yintercept(s) of the graph, if any exist.
 Follow the procedure in Example 1.2.3 to create a table of sample points on the graph of the
equation.  Plot the sample points and create a rough sketch of the graph of the equation.
 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 1.2.4
Exercise \(\PageIndex{41}\):
\( y=x^{2}+1\)
Exercise \(\PageIndex{42}\):
\( y=x^{2}\ \ 2x\ \ 8\)
Exercise \(\PageIndex{43}\):
\( y=x^{3}x\)
Exercise \(\PageIndex{44}\):
\( y=\frac{x^{3}}{4}3x\)
Exercise \(\PageIndex{45}\):
\( y=\sqrt{x2}\)
Exercise \(\PageIndex{46}\):
\( y=2\sqrt{x+4}2\)
Exercise \(\PageIndex{47}\):
\( 3x\ \ y=7\)
Exercise \(\PageIndex{48}\):
\( 3x2y=10\)
Exercise \(\PageIndex{49}\):
\( (x+2)^{2}+y^{2}=16\)
Exercise \(\PageIndex{50}\):
\( x^{2}y^{2}=1\)
Exercise \(\PageIndex{51}\):
\( y=x^{2}+1\)
Exercise \(\PageIndex{52}\):
\( x^{3}y=4\)
The procedures which we have outlined in the Examples of this section and used in Exercises 41  52 all rely on the fact that the equations were "wellbehaved". 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 53  56. 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 here.
Exercise \(\PageIndex{53}\):
Folium of Descartes:
\( x^{3}+y^{3}3xy\ =\ 0\)
Exercise \(\PageIndex{54}\):
Kampyle of Eudoxus:
\( x^{4}\ =\ x^{2}+y^{2}\)
Exercise \(\PageIndex{55}\):
Tschirnhausen cubic:
\( y^{2}\ =\ x^{3}+3x^{2}\)
Exercise \(\PageIndex{56}\):
Cooked egg:
\( (x^{2}+y^{2})^{2}\ =\ x^{3}+y^{3}\)
Exercise \(\PageIndex{57}\):
With the help of your classmates, nd examples of equations whose graphs possess
 symmetry about the \(x\)axis only
 symmetry about the \(y\)axis only
 symmetry about the origin only
 symmetry about the \(x\)axis, \(y\)axis, and origin
Can you nd an example of an equation whose graph possesses exactly two of the symmetries
listed above? Why or why not?
1.3 Introduction to Functions
In Exercises 1  12, determine whether or not the relation represents y as a function of x. Find the domain and range of those relations which are functions.
Exercise \(\PageIndex{1}\):
\( \{(3,9),\ (2,4),\ (1,1),\ (0,0),\ (1,1),\ (1,1),\ (1,1),\ (2,4),\ (3,9)\}\)
Exercise \(\PageIndex{2}\):
\( \{(3,0),\ (1,6),\ (2,3),\ (4,2),\ (5,6),\ (4,9),\ (6,2)\}\)
Exercise \(\PageIndex{3}\):
\( \{(3,0),\ (7,6),\ (5,5),\ (6,4),\ (4,9),\ (3,0)\}\)
Exercise \(\PageIndex{4}\):
\( \{(1,2),\ (4,4),\ (9,6),\ (16,8),\ (25,10),\ (36,12),...\}\)
Exercise \(\PageIndex{5}\):
\( \{(x,y)\ x \text{ is an odd integer, and }y\text{ is an even integer}\}\)
Exercise \(\PageIndex{6}\):
\( \{(x,1)\ x \text{ is an irrational number}\}\)
Exercise \(\PageIndex{7}\):
\( \{(1,0),\ (2,1),\ (4,2),\ (8,3),\ (16,4),\ (32,5),...\}\)
Exercise \(\PageIndex{8}\):
\(..., \{(3,9),\ (2,4),\ (1,1),\ (0,0),\ (1,1),\ (2,4),\ (3,9)...\}\)
Exercise \(\PageIndex{9}\):
\( \{ (2,y)3<y<4\}\)
Exercise \(\PageIndex{10}\):
\( \{ (x,3)2\leq x <4\}\)
Exercise \(\PageIndex{11}\):
\( \{(x,x^{2})\ x \text{ is an real number}\}\)
Exercise \(\PageIndex{12}\):
\( \{(x^{2},x)\ x \text{ is an real number}\}\)
In Exercises 13  32, determine whether or not the relation represents y as a function of x. Find the domain and range of those relations which are functions.
Exercise \(\PageIndex{13}\):
Exercise \(\PageIndex{14}\):
Exercise \(\PageIndex{15}\):
Exercise \(\PageIndex{16}\):
Exercise \(\PageIndex{17}\):
Exercise \(\PageIndex{18}\):
Exercise \(\PageIndex{19}\):
Exercise \(\PageIndex{20}\):
Exercise \(\PageIndex{21}\):
Exercise \(\PageIndex{22}\):
Exercise \(\PageIndex{23}\):
Exercise \(\PageIndex{24}\):
Exercise \(\PageIndex{25}\):
Exercise \(\PageIndex{26}\):
Exercise \(\PageIndex{27}\):
Exercise \(\PageIndex{28}\):
Exercise \(\PageIndex{29}\):
Exercise \(\PageIndex{30}\):
Exercise \(\PageIndex{31}\):
Exercise \(\PageIndex{32}\):
In Exercises 33  47, determine whether or not the equation represents \( y\) as a function of \( x\).
Exercise \(\PageIndex{33}\):
\( y\ =\ x^{3}\ \ x\)
Exercise \(\PageIndex{34}\):
\( y\ =\ \sqrt{x2}\)
Exercise \(\PageIndex{35}\):
\( x^{3}y\ =\ 4\)
Exercise \(\PageIndex{36}\):
\( x^{2}\ \ y^{2}\ =\ 1\)
Exercise \(\PageIndex{37}\):
\( y=\frac{x}{x^{2}9}\)
Exercise \(\PageIndex{38}\):
\( x\ =\ 6\)
Exercise \(\PageIndex{39}\):
\( x\ =\ y^{2}\ +\ 4\)
Exercise \(\PageIndex{40}\):
\( y\ =\ x^{2}\ +\ 4\)
Exercise \(\PageIndex{41}\):
\( x^{2}\ +\ y^{2}\ =\ 4\)
Exercise \(\PageIndex{42}\):
\( y\ =\ \sqrt{4x^{2}}\)
Exercise \(\PageIndex{43}\):
\( x^{2}\ \ y^{2}\ =\ 4\)
Exercise \(\PageIndex{44}\):
\( x^{3}\ +\ y^{3}\ =\ 4\)
Exercise \(\PageIndex{45}\):
\( 2x\ +\ 3y\ =\ 4\)
Exercise \(\PageIndex{46}\):
\( 2xy\ =\ 4\)
Exercise \(\PageIndex{47}\):
\( x^{2}\ =\ y^{2}\)
Exercise \(\PageIndex{48}\):
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?
Exercise \(\PageIndex{49}\):
Explain why the relation between your classmates and their email addresses may not be a function. What about phone numbers and Social Security Numbers?
The process given in Example 1.3.5 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 1.1. Discuss with your classmates how you might find such points for the relations given in Exercises 50  53.
Exercise \(\PageIndex{50}\):
\( x^{3}\ +\ y^{3}\ 3xy\ =\ 0\)
Exercise \(\PageIndex{51}\):
\( x^{4}\ =\ x^{2}\ +\ y^{2}\)
Exercise \(\PageIndex{52}\):
\( y^{2}\ =\ x^{3}\ +\ 3x^{2}\)
Exercise \(\PageIndex{53}\):
\( (x^{2}+y^{2})^{2}\ =\ x^{3}\ +\ y^{3}\)
1.4 Function Notation:
Exercise \(\PageIndex{1}\):
\( f\) is a function that takes a real number \( x\) and performs the following three steps in the order given:
(1) multiply by 2; (2) add 3; (3) divide by 4.
Exercise \(\PageIndex{2}\):
\( f\) 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.
Exercise \(\PageIndex{3}\):
\( f\) 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.
Exercise \(\PageIndex{4}\):
\( f\) is a function that takes a real number \( x\) and performs the following three steps in the order given:
(1) multiply by 2; (2) add 3; (3) take the square root.
Exercise \(\PageIndex{5}\):
\( f\) 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) take the square root.
Exercise \(\PageIndex{6}\):
\( f\) 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.
Exercise \(\PageIndex{7}\):
\( f\) 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.
Exercise \(\PageIndex{8}\):
\( f\) 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.
Exercise \(\PageIndex{9}\):
\( f\) 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.
Exercise \(\PageIndex{10}\):
\( f\) 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.
In Exercises 11  18, use the given function \( f\) to find and simplify the following:



Exercise \(\PageIndex{11}\):
\( f(x)\ =\ 2x\ +\ 1\)
Exercise \(\PageIndex{12}\):
\( f(x)\ =\ 3\ \ 4x\)
Exercise \(\PageIndex{13}\):
\( f(x)\ =\ 2\ \ x^{2}\)
Exercise \(\PageIndex{14}\):
\( f(x)\ =\ x^{2}\ \ 3x\ +\ 2\)
Exercise \(\PageIndex{15}\):
\( f(x)\ =\ \frac{x}{x1}\)
Exercise \(\PageIndex{16}\):
\( f(x)\ =\ \frac{2}{x^{3}}\)
Exercise \(\PageIndex{17}\):
\( f(x)\ =\ 6\)
Exercise \(\PageIndex{18}\):
\( f(x)\ = \ 0\)
In Exercises 19  26, use the given function \( f\) to find and simplify the following:



Exercise \(\PageIndex{19}\):
\( f(x)\ =\ 2x\ \ 5\)
Exercise \(\PageIndex{20}\):
\( f(x)\ =\ 5\ \ 2x\)
Exercise \(\PageIndex{21}\):
\( f(x)\ =\ 2x^{2}\ \ 1\)
Exercise \(\PageIndex{22}\):
\( f(x)\ =\ 3x^{2}\ +\ 3x\ \ 2\)
Exercise \(\PageIndex{23}\):
\( f(x)\ = \ \sqrt{2x\ +\ 1}\)
Exercise \(\PageIndex{24}\):
\( f(x)\ =\ 117}\)
Exercise \(\PageIndex{25}\):
\( f(x)\ =\ \frac{x}{2}\)
Exercise \(\PageIndex{26}\):
\( f(x)\ = \ \frac{2}{x}\)
In Exercises 27  34, use the given function \( f\) to find \( f(0)\) and solve \( f(x) = 0\)
Exercise \(\PageIndex{27}\):
\( f(x)\ =\ 2x\ \ 1\)
Exercise \(\PageIndex{28}\):
\( f(x)\ =\ 3\ \ \frac{2}{5}x\)
Exercise \(\PageIndex{29}\):
\( f(x)\ =\ 2x^{2}\ \ 6\)
Exercise \(\PageIndex{30}\):
\( f(x)\ =\ x^{2}\ \ x\ \ 12\)
Exercise \(\PageIndex{31}\):
\( f(x)\ = \ \sqrt{x\ +\ 4}\)
Exercise \(\PageIndex{32}\):
\( f(x)\ = \ \sqrt{1\ \ 2x}\)
Exercise \(\PageIndex{33}\):
\( f(x)\ =\ \frac{3}{4\ \ x}\)
Exercise \(\PageIndex{34}\):
\( f(x)\ =\ \frac{3x^{2}\ \ 12x}{4\ \ x^{2}}\)
Exercise \(\PageIndex{35}\):
Let \( f(x)\ =\ \begin{cases}x+5 &\text{if}\qquad x \leq3\\\sqrt{9x^{2}} &\text{if}\qquad 3<x\leq3\\x+5 &\text{if}\qquad x>3\end{cases}\) Compute the following function values.
Exercise \(\PageIndex{a}\):
\( f(4)\)
Exercise \(\PageIndex{b}\):
\( f(3)\)
Exercise \(\PageIndex{c}\):
\( f(3)\)
Exercise \(\PageIndex{d}\):
\( f(3.001)\)
Exercise \(\PageIndex{e}\):
\( f(3.001)\)
Exercise \(\PageIndex{f}\):
\( f(2)\)
Exercise \(\PageIndex{36}\):
Let \( f(x)\ =\ \begin{cases}x^{2} &\text{if}\qquad x \leq1\\\sqrt{1x^{2}} &\text{if}\qquad 1<x\leq1\\x &\text{if}\qquad x>1\end{cases} \) Compute the following function values.
Exercise \(\PageIndex{a}\):
\( f(4)\)
Exercise \(\PageIndex{b}\):
\( f(3)\)
Exercise \(\PageIndex{c}\):
\( f(1)\)
Exercise \(\PageIndex{d}\):
\( f(0)\)
Exercise \(\PageIndex{e}\):
\( f(1)\)
Exercise \(\PageIndex{f}\):
\( f(0.999)\)
In Exercises 37  62, find the (implied) domain of the function.
Exercise \(\PageIndex{37}\):
\( f(x)\ =\ x^{4}\ \ 13x^{3}\ +\ 56x^{2}\ \ 19\)
Exercise \(\PageIndex{38}\):
\( f(x)\ =\ x^{2}\ +\ 4\)
Exercise \(\PageIndex{39}\):
\( f(x)\ =\ \frac{x\ \ 2}{x\ +\ 1}\)
Exercise \(\PageIndex{40}\):
\( f(x)\ =\ \frac{3x}{x^{2}\ +\ x\ \ 2}\)
Exercise \(\PageIndex{41}\):
\( f(x)\ =\ \frac{2x}{x^{3}\ +\ 3}\)
Exercise \(\PageIndex{42}\):
\( f(x)\ =\ \frac{2x}{x^{3}\ \ 3}\)
Exercise \(\PageIndex{43}\):
\( f(x)\ =\ \frac{x\ +\ 4}{x^{2}\ \ 36}\)
Exercise \(\PageIndex{44}\):
\( f(x)\ =\ \frac{x\ \ 2}{x\ +\ 2}\)
Exercise \(\PageIndex{45}\):
\( f(x)\ =\ \sqrt{3\ \ x}\)
Exercise \(\PageIndex{46}\):
\( f(x)\ =\ \sqrt{2x\ +\ 5}\)
Exercise \(\PageIndex{47}\):
\( f(x)\ =\ 9x\sqrt{x\ +\ 3}\)
Exercise \(\PageIndex{48}\):
\( f(x)\ =\ \frac{\sqrt{7\ \ x}}{x^{2}\ +\ 1}\)
Exercise \(\PageIndex{49}\):
\( f(x)\ =\ \sqrt{6x\ \ 2}\)
Exercise \(\PageIndex{50}\):
\( f(x)\ =\ \frac{6}{\sqrt{6x\ \ 2}}\)
Exercise \(\PageIndex{51}\):
\( f(x)\ =\ \sqrt[3]{6x\ \ 2}\)
Exercise \(\PageIndex{52}\):
\( f(x)\ =\ \frac{6}{4\ \ \sqrt{6x\ \ 2}}\)
Exercise \(\PageIndex{53}\):
\( f(x)\ =\ \frac{\sqrt{6x\ \ 2}}{x^{2}\ \ 36}\)
Exercise \(\PageIndex{54}\):
\( f(x)\ =\ \frac{\sqrt[3]{6x\ \ 2}}{x^{2}\ +\ 36}\)
Exercise \(\PageIndex{55}\):
\( s(t)\ =\ \frac{t}{t8}\)
Exercise \(\PageIndex{56}\):
\( Q(r)\ =\ \frac{\sqrt{r}}{r8}\)
Exercise \(\PageIndex{57}\):
\( b(\theta)=\frac{\theta}{\sqrt{\theta\ \ 8}}\)
Exercise \(\PageIndex{58}\):
\( A(x)\ =\ \sqrt{x\ \ 7}\ +\ \sqrt{9\ \ x}\)
Exercise \(\PageIndex{59}\):
\( \alpha(y)\ =\ \sqrt[3]{\frac{y}{y8}}\)
Exercise \(\PageIndex{60}\):
\( g(v)\ =\ \frac{1}{4\ \ \frac{1}{v^{2}}}\)
Exercise \(\PageIndex{61}\):
\( T(t)\ =\ \frac{\sqrt{t}\ \ 8}{5\ \ t}\)
Exercise \(\PageIndex{62}\):
\( u(w)\ =\ \frac{w\ \ 8}{5\ \ \sqrt{w}}\)
Exercise \(\PageIndex{63}\):
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 \( A(x)\ =\ x2\) for \( x\ >\ 0\). Find \( A(3)\) and solve \( A(x)\ =\ 36\). Interpret your answers to each. Why is \( x\) restricted to \(x\ >\ 0\)?
Exercise \(\PageIndex{64}\):
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 \(A(x)\ =\ x2\) for \( x\ >\ 0.\) Find \( A(3)\) and solve \( A(x)\ =\ 36\). Interpret your answers to each. Why is \( x\) restricted to \( x\ >\ 0\)?
Exercise \(\PageIndex{65}\):
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 \( V(x)\ =\ x3\) for \( x\ >\ 0\). Find \( V(5)\) and solve \( V(x)\ =\ 27\). Interpret your answers to each. Why is \( x\) restricted to \( x\ >\ 0\)?
Exercise \(\PageIndex{66}\):
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 \( r\ >\ 0\)?
Exercise \(\PageIndex{67}\):
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 \( 0\ \leq\ t\ \leq\ 2\)?
Exercise \(\PageIndex{68}\):
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 \( T(12)\).
Exercise \(\PageIndex{69}\):
The function \( C(x)\ =\ x^{2}\ \ 10x\ +\ 27\) models the cost, in hundreds of dollars, to produce \( x\) thousand pens. Find and interpret \( C(0),\ C(2)\) and \( C(5)\).
Exercise \(\PageIndex{70}\):
Using data from the 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.
Exercise \(\PageIndex{71}\):
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.
Exercise \(\PageIndex{72}\):
For \( n\) copies of the book \( Me and my Sasquatch\), a print ondemand company charges \( C(n)\) dollars, where \( C(n)\) is determined by the formula
\( C(n)=\begin{cases}15n & \text{if} & 1\leq n\leq25 \\13.50n & \text{if} & 25<n\leq50 \\ 12n & \text{if} & n>50\end{cases}\)
Exercise \(\PageIndex{A}\):
Find and interpret \( C(20)\).
Exercise \(\PageIndex{B}\):
How much does it cost to order 50 copies of the book? What about 51 copies?
Exercise \(\PageIndex{C}\):
Your answer to 72b 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.)
Exercise \(\PageIndex{73}\):
An online comic book retailer charges shipping costs according to the following formula
\( S(n)=\begin{cases}1.5n+2.5 & \text{if} & 1\leq n\leq14 \\0 & \text{if} & n\geq15\end{cases}\)
where n is the number of comic books purchased and S(n) is the shipping cost in dollars.
Exercise \(\PageIndex{A}\):
What is the cost to ship 10 comic books?
Exercise \(\PageIndex{b}\):
What is the signicance of the formula \( S(n) = 0\) for \( n \geq 15\)?
Exercise \(\PageIndex{74}\):
The cost \( C\) (in dollars) to talk m minutes a month on a mobile phone plan is modeled by
\( C(m)=\begin{cases}25 & \text{if} & 0\leq m\leq1000 \\25\ +\ 0.1(m\ \ 1000) & \text{if} & m>1000\end{cases} \)
Exercise \(\PageIndex{a}\):
How much does it cost to talk 750 minutes per month with this plan?
Exercise \(\PageIndex{b}\):
How much does it cost to talk 750 minutes per month with this plan?
Exercise \(\PageIndex{75}\):
In Section 1.1.1 we dened the set of integers as \( Z\ =\ \{...\ 3,\ 2,\ 1,\ 0,\ 1,\ 2,\ 3,...\}\) The greatest integer of x, denoted by \( \llcorner x \lrcorner\), is defined to be the largest integer \( k\) with \( k \leq x\)
Exercise \(\PageIndex{A}\):
Find \( \llcorner0.785 \lrcorner,\ \llcorner117 \lrcorner,\ \llcorner2.001\lrcorner,\ \text{and } \llcorner\pi+6\lrcorner\)
Exercise \(\PageIndex{B}\):
Discuss with your classmates how \( \llcorner x \lrcorner\) may be described as a piecewise defined function.
HINT: There are infinitely many pieces!
Exercise \(\PageIndex{C}\):
Is \( \llcorner a+b\lrcorner\ =\ \llcorner a \lrcorner+\llcorner b \lrcorner\) always true? What if \( a\) or \( b\) is an integer? Test some values, make a conjecture, and explain your result.
Exercise \(\PageIndex{76}\):
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.
Exercise \(\PageIndex{A}\):
\( f(0)\ =\ f(1\ +\ 1)\ =\ f(1)\ +\ f(1)\)
Exercise \(\PageIndex{B}\):
\( f(5)\ =\ f(2 + 3)\ =\ f(2)\ +\ f(3)\)
Exercise \(\PageIndex{c}\):
\( f(6)\ =\ f(0\ \ 6)\ =\ f(0)\ \ f(6)\)
Exercise \(\PageIndex{D}\):
\( f(a\ +\ b)\ =\ f(a)\ +\ f(b)\)
How many functions did you find that failed to satisfy the conditions above? Did f(x) = x2 work? What about \( f(x)\ =\ \sqrt{x} \) or \( f(x)\ =\ 3x\ +\ 7\) or \(f(x)\ =\ \frac{1}{x}\) ? 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, in general, \( f(a\ +\ b)\ \neq\ f(a)\ +\ f(b)\).
1.5 Function Arithmetic:
In Exercises 1  10, use the pair of functions f and g to nd the following values if they exist.






Exercise \(\PageIndex{1}\):
\( f(x)\ =\ 3x\ +\ 1\) and \(g(x)\ =\ 4\ \ x\)
Exercise \(\PageIndex{2}\):
\( f(x)\ =\ x^{2}\) and \(g(x)\ =\ 2x\ +\ 1\)
Exercise \(\PageIndex{3}\):
\( f(x)\ =\ x^{2}\ \ x\) and \(g(x)\ =\ 12\ \ x^{2}\)
Exercise \(\PageIndex{4}\):
\( f(x)\ =\ 2x^{3}\) and \(g(x)\ =\ x^{2}\ \ 2x\ \ 3\)
Exercise \(\PageIndex{5}\):
\( f(x)\ =\ sqrt{x+3}\) and \(g(x)\ =\ 2x\ \ 1\)
Exercise \(\PageIndex{6}\):
\( f(x)\ =\ sqrt{4x}\) and \(g(x)\ =\ sqrt{x+2}\)
Exercise \(\PageIndex{7}\):
\( f(x)\ =\ 2x\) and \(g(x)\ =\ \frac{1}{2x\ +\ 1}\)
Exercise \(\PageIndex{8}\):
\( f(x)\ =\ x^{2}\) and \(g(x)\ =\ \frac{3}{2x\ \ 3}\)
Exercise \(\PageIndex{9}\):
\( f(x)\ =\ x^{2}\) and \(g(x)\ =\ \frac{1}{x^{2}}\)
Exercise \(\PageIndex{10}\):
\( f(x)\ =\ x^{2}\ +\ 1\) and \(g(x)\ =\ \frac{1}{x^{2}\ +\ 1}\)
In Exercises 11  20, use the pair of functions \( f\) and \( g\) to find the domain of the indicated function then find and simplify an expression for it.




Exercise \(\PageIndex{11}\):
\( f(x)\ =\ 2x\ +\ 1\) and \(g(x)\ =\ x\ \ 2\)
Exercise \(\PageIndex{12}\):
\( f(x)\ =\ 1\ \ 4x\) and \(g(x)\ =\ 2x\ \ 1\)
Exercise \(\PageIndex{13}\):
\( f(x)\ =\ x^{2}\) and \(g(x)\ =\ 3x\ \ 1\)
Exercise \(\PageIndex{14}\):
\( f(x)\ =\ x^{2}\ \ x\) and \(g(x)\ =\ 7x\)
Exercise \(\PageIndex{15}\):
\( f(x)\ =\ x^{2}\ \ 4\) and \(g(x)\ =\ 3x\ +\ 6\)
Exercise \(\PageIndex{16}\):
\( f(x)\ =\ x^{2}\ +\ x\ +\ 6\) and \(g(x)\ =\ x^{2}\ \ 9\)
Exercise \(\PageIndex{17}\):
\( f(x)\ =\ \frac{x}{2}\) and \(g(x)\ =\ \frac{2}{x}\)
Exercise \(\PageIndex{18}\):
\( f(x)\ =\ x\ \ 1\) and \(g(x)\ =\ \frac{1}{x\ \ 1}\)
Exercise \(\PageIndex{19}\):
\( f(x)\ =\ x\) and \(g(x)\ =\ \sqrt{x\ +\ 1}\)
Exercise \(\PageIndex{20}\):
\( f(x)\ =\ \sqrt{x\ \ 5}\) and \(g(x)\ =\ f(x)\ =\ \sqrt{x\ \ 5}\)
In Exercises 21  45, find and simplify the difference quotient \( \frac{f(x + h)  f(x)}{h}\) for the given function.
Exercise \(\PageIndex{21}\):
\( f(x)\ =\ 2x\ \ 5\)
Exercise \(\PageIndex{22}\):
\( f(x)\ =\ 3x\ +\ 5\)
Exercise \(\PageIndex{23}\):
\( f(x)\ =\ 6\)
Exercise \(\PageIndex{24}\):
\( f(x)\ =\ 3x^{2}\ \ x\)
Exercise \(\PageIndex{25}\):
\( f(x)\ =\ x^{2}\ +\ 2x\ \ 1\)
Exercise \(\PageIndex{26}\):
\( f(x)\ =\ 4x^{2} \)
Exercise \(\PageIndex{27}\):
\( f(x)\ =\ x\ \ x^{2}\)
Exercise \(\PageIndex{28}\):
\( f(x)\ =\ x^{3}\ +\ 1\)
Exercise \(\PageIndex{29}\):
\( f(x)\ =\ mx\ +\ b\ \text{where } m \dne 0\)
Exercise \(\PageIndex{30}\):
\( f(x)\ =\ ax^{2}\ +\ bx\ +\ c\ \text{where } a \dne 0\)
Exercise \(\PageIndex{31}\):
\( f(x)\ =\ \frac{2}{x}\)
Exercise \(\PageIndex{32}\):
\( f(x)\ =\ \frac{3}{1\ \ x}\)
Exercise \(\PageIndex{33}\):
\( f(x)\ =\ \frac{1}{x^{2}}\)
Exercise \(\PageIndex{34}\):
\( f(x)\ =\ \frac{2}{x\ +\ 5}\)
Exercise \(\PageIndex{35}\):
\( f(x)\ =\ \frac{1}{4x\ \ 3}\)
Exercise \(\PageIndex{36}\):
\( f(x)\ =\ \frac{3x}{x\ +\ 1}\)
Exercise \(\PageIndex{37}\):
\( f(x)\ =\ \frac{x}{x\ \ 9}\)
Exercise \(\PageIndex{38}\):
\( f(x)\ =\ \frac{x^{2}}{2x\ +\ 1}\)
Exercise \(\PageIndex{39}\):
\( f(x)\ =\ \sqrt{x\ \ 9}\)
Exercise \(\PageIndex{40}\):
\( f(x)\ =\ \sqrt{2x\ +\ 1}\)
Exercise \(\PageIndex{41}\):
\( f(x)\ =\ \sqrt{4x\ +\ 5}\)
Exercise \(\PageIndex{42}\):
\( f(x)\ =\ \sqrt{4\ \ x}\)
Exercise \(\PageIndex{43}\):
\( f(x)\ =\ \sqrt{ax\ +\ b}\ \text{where } a \dne 0\)
Exercise \(\PageIndex{44}\):
\( f(x)\ =\ x\sqrt{x}\)
Exercise \(\PageIndex{45}\):
\( f(x)\ =\ \sqrt[3]{x}\)
HINT: \( (a\ \ b)(a^{2}\ +\ ab\ +\ b^{2})\ =\ a^{3}b^{3}\)
In Exercises 46  50, \( C(x)\) denotes the cost to produce \( x\) items and \( p(x)\) denotes the pricedemand function in the given economic scenario. In each Exercise, do the following:






Exercise \(\PageIndex{46}\):
The cost, in dollars, to produce \( x\) "I'd rather be a Sasquatch" TShirts is \( C(x)\ =\ 2x\ +\ 26\), \(x\ \geq\ 0\) and the pricedemand function, in dollars per shirt, is \( p(x)\ =\ 90\ \ 3x,\ 0\leq x\leq 30\).
Exercise \(\PageIndex{47}\):
The cost, in dollars, to produce \( x\) bottles of 100% AllNatural Certied FreeTrade Organic Sasquatch Tonic is \( C(x)\ =\ 10x\ +\ 100, \ x\geq 0\) and the pricedemand function, in dollars per bottle, is \( p(x)\ =\ 90\ \ 3x,\ 0\leq x\leq 30\).
Exercise \(\PageIndex{48}\):
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 pricedemand function, in cents per cup, is \( p(x)\ =\ 90\ \ 3x\), \(0\leq x \leq 30\).
Exercise \(\PageIndex{49}\):
The daily cost, in dollars, to produce \( x\) Sasquatch Berry Pies \( C(x)\ =\ 3x\ +\ 36\),\( x\geq 0\) and the pricedemand function, in dollars per pie, is \( p(x)\ =\ 12\ \ 0.5x\), \( 0\leq x\leq 24\).
Exercise \(\PageIndex{50}\):
The monthly cost, in hundreds of dollars, to produce \( x\) custom built electric scooters is \( C(x)\ =\ 20x\ +\ 1000\), \( x \geq 0\) and the pricedemand function, in hundreds of dollars per scooter, is \( p(x)\ =\ 140\ \ 2x\), \( 0\leq x\leq 70\).
In Exercises 51  62, let \( f\) be the function de 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.
Exercise \(\PageIndex{51}\):
\( (f\ +\ g)(3)\)
Exercise \(\PageIndex{52}\):
\( (f\ \ g)(2)\)
Exercise \(\PageIndex{53}\):
\( (fg)(1)\)
Exercise \(\PageIndex{54}\):
\( (g\ +\ f)(1)\)
Exercise \(\PageIndex{55}\):
\( (g\ \ f)(3)\)
Exercise \(\PageIndex{56}\):
\( (gf)(3)\)
Exercise \(\PageIndex{57}\):
\( (\frac{f}{g})(2)\)
Exercise \(\PageIndex{58}\):
\( (\frac{f}{g})(1)\)
Exercise \(\PageIndex{59}\):
\( (\frac{f}{g})(2)\)
Exercise \(\PageIndex{60}\):
\( (\frac{g}{f})(1)\)
Exercise \(\PageIndex{61}\):
\( (\frac{g}{f})(3)\)
Exercise \(\PageIndex{62}\):
\( (\frac{g}{f})(3)\)