Stewart Precalc 6e Section 2.2: Graphs of Functions

1 Exercises - Concepts · Level 1

To graph the function $f$, we plot the points $(x, \text{___})$ in a coordinate plane. To graph $f(x) = x^3 + 2$, we plot the points $(x, \text{___})$. So the point $(2, \text{___})$ is on the graph of $f$. The height of the graph of $f$ above the $x$-axis when $x = 2$ is ___.

2 Exercises - Concepts · Level 1

If $f(2) = 3$, then the point $(2, \text{___})$ is on the graph of $f$.

3 Exercises - Concepts · Level 1

If the point $(2, 3)$ is on the graph of $f$, then $f(2) = $ ___.

4 Exercises - Concepts · Level 1

Match the function with its graph.

(a) $f(x) = x^2$

Answer for 4(a)

(b) $f(x) = x^3$

Answer for 4(b)

(c) $f(x) = \sqrt{x}$

Answer for 4(c)

(d) $f(x) = |x|$

Answer for 4(d)

Enter your answer directly below each part above.

5 Exercises - Sketch by Table of Values · Level 1

Sketch the graph of the function by first making a table of values: $f(x) = 2$.

6 Exercises - Sketch by Table of Values · Level 1

Sketch the graph of the function by first making a table of values: $f(x) = -3$.

7 Exercises - Sketch by Table of Values · Level 1

Sketch the graph of the function by first making a table of values: $f(x) = 2x - 4$.

8 Exercises - Sketch by Table of Values · Level 1

Sketch the graph of the function by first making a table of values: $f(x) = 6 - 3x$.

9 Exercises - Sketch by Table of Values · Level 1

Sketch the graph of the function by first making a table of values: $f(x) = -x + 3$, $-3 \leq x \leq 3$.

10 Exercises - Sketch by Table of Values · Level 1

Sketch the graph of the function by first making a table of values: $f(x) = \dfrac{x - 3}{2}$, $0 \leq x \leq 5$.

11 Exercises - Sketch by Table of Values · Level 1

Sketch the graph of the function by first making a table of values: $f(x) = -x^2$.

12 Exercises - Sketch by Table of Values · Level 1

Sketch the graph of the function by first making a table of values: $f(x) = x^2 - 4$.

13 Exercises - Sketch by Table of Values · Level 1

Sketch the graph of the function by first making a table of values: $h(x) = 16 - x^2$.

14 Exercises - Sketch by Table of Values · Level 2

Sketch the graph of the function by first making a table of values: $g(x) = (x - 3)^2$.

15 Exercises - Sketch by Table of Values · Level 2

Sketch the graph of the function by first making a table of values: $g(x) = x^3 - 8$.

16 Exercises - Sketch by Table of Values · Level 2

Sketch the graph of the function by first making a table of values: $q(x) = (x + 2)^3$.

17 Exercises - Sketch by Table of Values · Level 2

Sketch the graph of the function by first making a table of values: $g(x) = x^2 - 2x$.

18 Exercises - Sketch by Table of Values · Level 3

Sketch the graph of the function by first making a table of values: $h(x) = 4x^2 - x^4$.

19 Exercises - Sketch by Table of Values · Level 2

Sketch the graph of the function by first making a table of values: $f(x) = 1 + \sqrt{x}$.

20 Exercises - Sketch by Table of Values · Level 2

Sketch the graph of the function by first making a table of values: $f(x) = \sqrt{x + 4}$.

21 Exercises - Sketch by Table of Values · Level 2

Sketch the graph of the function by first making a table of values: $g(x) = -\sqrt{x}$.

22 Exercises - Sketch by Table of Values · Level 2

Sketch the graph of the function by first making a table of values: $g(x) = \sqrt{-x}$.

23 Exercises - Sketch by Table of Values · Level 2

Sketch the graph of the function by first making a table of values: $H(x) = |2x|$.

24 Exercises - Sketch by Table of Values · Level 2

Sketch the graph of the function by first making a table of values: $H(x) = |x + 1|$.

25 Exercises - Sketch by Table of Values · Level 2

Sketch the graph of the function by first making a table of values: $G(x) = |x| + x$.

26 Exercises - Sketch by Table of Values · Level 2

Sketch the graph of the function by first making a table of values: $G(x) = |x| - x$.

27 Exercises - Sketch by Table of Values · Level 2

Sketch the graph of the function by first making a table of values: $f(x) = |2x - 2|$.

28 Exercises - Sketch by Table of Values · Level 2

Sketch the graph of the function by first making a table of values: $f(x) = \dfrac{x}{|x|}$.

29 Exercises - Viewing Rectangle · Level 2

Graph the function $f(x) = 8x - x^2$ in each of the given viewing rectangles, and select the one that produces the most appropriate graph. (a) $[-5, 5]$ by $[-5, 5]$ (b) $[-10, 10]$ by $[-10, 10]$ (c) $[-2, 10]$ by $[-5, 20]$ (d) $[-10, 10]$ by $[-100, 100]$

A

(a) $[-5, 5]$ by $[-5, 5]$

B

(b) $[-10, 10]$ by $[-10, 10]$

C

(c) $[-2, 10]$ by $[-5, 20]$

D

(d) $[-10, 10]$ by $[-100, 100]$

30 Exercises - Viewing Rectangle · Level 2

Graph the function $g(x) = x^2 - x - 20$ in each of the given viewing rectangles, and select the one that produces the most appropriate graph. (a) $[-2, 2]$ by $[-5, 5]$ (b) $[-10, 10]$ by $[-10, 10]$ (c) $[-7, 7]$ by $[-25, 20]$ (d) $[-10, 10]$ by $[-100, 100]$

A

(a) $[-2, 2]$ by $[-5, 5]$

B

(b) $[-10, 10]$ by $[-10, 10]$

C

(c) $[-7, 7]$ by $[-25, 20]$

D

(d) $[-10, 10]$ by $[-100, 100]$

31 Exercises - Viewing Rectangle · Level 2

Graph the function $h(x) = x^3 - 5x - 4$ in each of the given viewing rectangles, and select the one that produces the most appropriate graph. (a) $[-2, 2]$ by $[-2, 2]$ (b) $[-3, 3]$ by $[-10, 10]$ (c) $[-3, 3]$ by $[-10, 5]$ (d) $[-10, 10]$ by $[-10, 10]$

A

(a) $[-2, 2]$ by $[-2, 2]$

B

(b) $[-3, 3]$ by $[-10, 10]$

C

(c) $[-3, 3]$ by $[-10, 5]$

D

(d) $[-10, 10]$ by $[-10, 10]$

32 Exercises - Viewing Rectangle · Level 2

Graph the function $k(x) = \dfrac{1}{32} x^4 - x^2 + 2$ in each of the given viewing rectangles, and select the one that produces the most appropriate graph. (a) $[-1, 1]$ by $[-1, 1]$ (b) $[-2, 2]$ by $[-2, 2]$ (c) $[-5, 5]$ by $[-5, 5]$ (d) $[-10, 10]$ by $[-10, 10]$

A

(a) $[-1, 1]$ by $[-1, 1]$

B

(b) $[-2, 2]$ by $[-2, 2]$

C

(c) $[-5, 5]$ by $[-5, 5]$

D

(d) $[-10, 10]$ by $[-10, 10]$

33 Exercises - Piecewise Functions · Level 2

Sketch the graph of the piecewise defined function: $f(x) = \begin{cases} 0 & \quad \text{if } x < 2 \\ 1 & \quad \text{if } x \geq 2 \end{cases}$.

34 Exercises - Piecewise Functions · Level 2

Sketch the graph of the piecewise defined function: $f(x) = \begin{cases} 1 & \quad \text{if } x \leq 1 \\ x + 1 & \quad \text{if } x > 1 \end{cases}$.

35 Exercises - Piecewise Functions · Level 2

Sketch the graph of the piecewise defined function: $f(x) = \begin{cases} 3 & \quad \text{if } x < 2 \\ x - 1 & \quad \text{if } x \geq 2 \end{cases}$.

36 Exercises - Piecewise Functions · Level 2

Sketch the graph of the piecewise defined function: $f(x) = \begin{cases} 1 - x & \quad \text{if } x < -2 \\ 5 & \quad \text{if } x \geq -2 \end{cases}$.

37 Exercises - Piecewise Functions · Level 2

Sketch the graph of the piecewise defined function: $f(x) = \begin{cases} x & \quad \text{if } x \leq 0 \\ x + 1 & \quad \text{if } x > 0 \end{cases}$.

38 Exercises - Piecewise Functions · Level 2

Sketch the graph of the piecewise defined function: $f(x) = \begin{cases} 2x + 3 & \quad \text{if } x < -1 \\ 3 - x & \quad \text{if } x \geq -1 \end{cases}$.

39 Exercises - Piecewise Functions · Level 2

Sketch the graph of the piecewise defined function: $f(x) = \begin{cases} -1 & \quad \text{if } x < -1 \\ 1 & \quad \text{if } -1 \leq x \leq 1 \\ -1 & \quad \text{if } x > 1 \end{cases}$.

40 Exercises - Piecewise Functions · Level 2

Sketch the graph of the piecewise defined function: $f(x) = \begin{cases} -1 & \quad \text{if } x < -1 \\ x & \quad \text{if } -1 \leq x \leq 1 \\ 1 & \quad \text{if } x > 1 \end{cases}$.

41 Exercises - Piecewise Functions · Level 2

Sketch the graph of the piecewise defined function: $f(x) = \begin{cases} 2 & \quad \text{if } x \leq -1 \\ x^2 & \quad \text{if } x > -1 \end{cases}$.

42 Exercises - Piecewise Functions · Level 2

Sketch the graph of the piecewise defined function: $f(x) = \begin{cases} 1 - x^2 & \quad \text{if } x \leq 2 \\ x & \quad \text{if } x > 2 \end{cases}$.

43 Exercises - Piecewise Functions · Level 2

Sketch the graph of the piecewise defined function: $f(x) = \begin{cases} 0 & \quad \text{if } |x| \leq 2 \\ 3 & \quad \text{if } |x| > 2 \end{cases}$.

44 Exercises - Piecewise Functions · Level 2

Sketch the graph of the piecewise defined function: $f(x) = \begin{cases} x^2 & \quad \text{if } |x| \leq 1 \\ 1 & \quad \text{if } |x| > 1 \end{cases}$.

45 Exercises - Piecewise Functions · Level 3

Sketch the graph of the piecewise defined function: $f(x) = \begin{cases} 4 & \quad \text{if } x < -2 \\ x^2 & \quad \text{if } -2 \leq x \leq 2 \\ -x + 6 & \quad \text{if } x > 2 \end{cases}$.

46 Exercises - Piecewise Functions · Level 3

Sketch the graph of the piecewise defined function: $f(x) = \begin{cases} -x & \quad \text{if } x \leq 0 \\ 9 - x^2 & \quad \text{if } 0 < x \leq 3 \\ x - 3 & \quad \text{if } x > 3 \end{cases}$.

47 Exercise - Piecewise Functions · Level 2

Use a graphing device to draw the graph of the piecewise defined function. $f(x) = \begin{cases} x+2 & \quad \text{if } x \leq -1 \\ x^2 & \quad \text{if } x > -1 \end{cases}$

48 Exercise - Piecewise Functions · Level 2

Use a graphing device to draw the graph of the piecewise defined function. $f(x) = \begin{cases} 2x - x^2 & \quad \text{if } x > 1 \\ (x-1)^3 & \quad \text{if } x \leq 1 \end{cases}$

49 Exercise - Finding Piecewise Formula from Graph · Level 3

The graph of a piecewise defined function is given. Find a formula for the function in the indicated form: $f(x) = \begin{cases} \text{___} & \quad \text{if } x < -2 \\ \text{___} & \quad \text{if } -2 \leq x \leq 2 \\ \text{___} & \quad \text{if } x > 2 \end{cases}$

50 Exercise - Finding Piecewise Formula from Graph · Level 3

The graph of a piecewise defined function is given. Find a formula for the function in the indicated form: $f(x) = \begin{cases} \text{___} & \quad \text{if } x \leq -1 \\ \text{___} & \quad \text{if } -1 < x \leq 2 \\ \text{___} & \quad \text{if } x > 2 \end{cases}$

51 Exercise - Vertical Line Test · Level 2

Use the Vertical Line Test to determine whether each of the curves shown in parts (a), (b), (c), and (d) is the graph of a function of $x$.

52 Exercise - Vertical Line Test · Level 2

Use the Vertical Line Test to determine whether each of the curves shown in parts (a), (b), (c), and (d) is the graph of a function of $x$.

53 Exercise - Vertical Line Test with Domain and Range · Level 3

Use the Vertical Line Test to determine whether the curve is the graph of a function of $x$. If it is, state the domain and range of the function.

54 Exercise - Vertical Line Test with Domain and Range · Level 3

Use the Vertical Line Test to determine whether the curve is the graph of a function of $x$. If it is, state the domain and range of the function.

55 Exercise - Vertical Line Test with Domain and Range · Level 3

Use the Vertical Line Test to determine whether the curve is the graph of a function of $x$. If it is, state the domain and range of the function.

56 Exercise - Vertical Line Test with Domain and Range · Level 3

Use the Vertical Line Test to determine whether the curve is the graph of a function of $x$. If it is, state the domain and range of the function.

57 Exercise - Functions Defined by Equations · Level 1

Determine whether the equation defines $y$ as a function of $x$. $x^2 + 2y = 4$

58 Exercise - Functions Defined by Equations · Level 1

Determine whether the equation defines $y$ as a function of $x$. $3x + 7y = 21$

59 Exercise - Functions Defined by Equations · Level 1

Determine whether the equation defines $y$ as a function of $x$. $x = y^2$

60 Exercise - Functions Defined by Equations · Level 1

Determine whether the equation defines $y$ as a function of $x$. $x^2 + (y - 1)^2 = 4$

61 Exercise - Functions Defined by Equations · Level 1

Determine whether the equation defines $y$ as a function of $x$. $x + y^2 = 9$

62 Exercise - Functions Defined by Equations · Level 1

Determine whether the equation defines $y$ as a function of $x$. $x^2 + y = 9$

63 Exercise - Functions Defined by Equations · Level 2

Determine whether the equation defines $y$ as a function of $x$. $x^2 y + y = 1$

64 Exercise - Functions Defined by Equations · Level 2

Determine whether the equation defines $y$ as a function of $x$. $\sqrt{x} + y = 12$

65 Exercise - Functions Defined by Equations · Level 2

Determine whether the equation defines $y$ as a function of $x$. $2 |x| + y = 0$

66 Exercise - Functions Defined by Equations · Level 2

Determine whether the equation defines $y$ as a function of $x$. $2x + |y| = 0$

67 Exercise - Functions Defined by Equations · Level 2

Determine whether the equation defines $y$ as a function of $x$. $x = y^3$

68 Exercise - Functions Defined by Equations · Level 2

Determine whether the equation defines $y$ as a function of $x$. $x = y^4$

69 Exercise - Family of Functions · Level 2

A family of functions $f(x) = x^2 + c$ is given. (a) Graph all the members for $c = 0, 2, 4, 6$ in the viewing rectangle $[-5, 5]$ by $[-10, 10]$. (b) Graph all the members for $c = 0, -2, -4, -6$ in the viewing rectangle $[-5, 5]$ by $[-10, 10]$. (c) How does the value of $c$ affect the graph?

70 Exercise - Family of Functions · Level 2

A family of functions $f(x) = (x - c)^2$ is given. (a) Graph all the members for $c = 0, 1, 2, 3$ in the viewing rectangle $[-5, 5]$ by $[-10, 10]$. (b) Graph all the members for $c = 0, -1, -2, -3$ in the viewing rectangle $[-5, 5]$ by $[-10, 10]$. (c) How does the value of $c$ affect the graph?

71 Exercise - Family of Functions · Level 2

A family of functions $f(x) = (x - c)^3$ is given. (a) Graph all the members for $c = 0, 2, 4, 6$ in the viewing rectangle $[-10, 10]$ by $[-10, 10]$. (b) Graph all the members for $c = 0, -2, -4, -6$ in the viewing rectangle $[-10, 10]$ by $[-10, 10]$. (c) How does the value of $c$ affect the graph?

72 Exercise - Family of Functions · Level 2

A family of functions $f(x) = c x^2$ is given. (a) Graph all the members for $c = 1, \dfrac{1}{2}, 2, 4$ in the viewing rectangle $[-5, 5]$ by $[-10, 10]$. (b) Graph all the members for $c = 1, -1, -\dfrac{1}{2}, -2$ in the viewing rectangle $[-5, 5]$ by $[-10, 10]$. (c) How does the value of $c$ affect the graph?

73 Exercise - Family of Functions · Level 3

A family of functions $f(x) = x^c$ is given. (a) Graph all the members for $c = \dfrac{1}{2}, \dfrac{1}{4}, \dfrac{1}{6}$ in the viewing rectangle $[-1, 4]$ by $[-1, 3]$. (b) Graph all the members for $c = 1, \dfrac{1}{3}, \dfrac{1}{5}$ in the viewing rectangle $[-3, 3]$ by $[-2, 2]$. (c) How does the value of $c$ affect the graph?

74 Exercise - Family of Functions · Level 3

A family of functions $f(x) = x^n$ is given. (a) Graph the members for $n = 1, 3$ in the viewing rectangle $[-3, 3]$ by $[-3, 3]$. (b) Graph the members for $n = 2, 4$ in the viewing rectangle $[-3, 3]$ by $[-3, 3]$. (c) How does the value of $n$ affect the graph?

75 Exercise - Finding a Function from a Curve · Level 2

Find a function whose graph is the line segment joining the points $(-2, 1)$ and $(4, -6)$.

76 Exercise - Finding a Function from a Curve · Level 2

Find a function whose graph is the line segment joining the points $(-3, -2)$ and $(6, 3)$.

77 Exercise - Finding a Function from a Curve · Level 2

Find a function whose graph is the top half of the circle $x^2 + y^2 = 9$.

78 Exercise - Finding a Function from a Curve · Level 2

Find a function whose graph is the bottom half of the circle $x^2 + y^2 = 9$.

79 Application - Weather Balloon · Level 2

*Weather Balloon.* As a weather balloon is inflated, the thickness $T$ of its rubber skin is related to the radius of the balloon by $T(r) = \dfrac{0.5}{r^2}$, where $T$ and $r$ are measured in centimeters. Graph the function $T$ for values of $r$ between $10$ and $100$.

80 Application - Wind Turbine Power · Level 2

*Power from a Wind Turbine.* The power produced by a wind turbine depends on the speed of the wind. If a windmill has blades 3 meters long, then the power $P$ produced by the turbine is modeled by $P(v) = 14.1 v^3$, where $P$ is measured in watts (W) and $v$ is measured in meters per second (m/s). Graph the function $P$ for wind speeds between 1 m/s and 10 m/s.

81 Application - Utility Rates Piecewise · Level 3

*Utility Rates.* Westside Energy charges its electric customers a base rate of \$6.00 per month, plus 10 cents per kilowatt-hour (kWh) for the first 300 kWh used and 6 cents per kWh for all usage over 300 kWh. Suppose a customer uses $x$ kWh of electricity in one month. (a) Express the monthly cost $E$ as a piecewise-defined function of $x$. (b) Graph the function $E$ for $0 \leq x \leq 600$.

82 Application - Taxicab Step Function · Level 3

*Taxicab Function.* A taxi company charges \$2.00 for the first mile (or part of a mile) and 20 cents for each succeeding tenth of a mile (or part). Express the cost $C$ (in dollars) of a ride as a piecewise-defined function of the distance $x$ traveled (in miles) for $0 < x < 2$, and sketch the graph of this function.

83 Application - Postage Step Function · Level 3

*Postage Rates.* The domestic postage rate for first-class letters weighing 3.5 oz or less is 44 cents for the first ounce (or less), plus 17 cents for each additional ounce (or part of an ounce). Express the postage $P$ as a piecewise-defined function of the weight $x$ of a letter, with $0 < x \leq 3.5$, and sketch the graph of this function.

84 Discussion - When Does a Graph Represent a Function · Level 3

*When Does a Graph Represent a Function?* For every integer $n$, the graph of the equation $y = x^n$ is the graph of a function, namely $f(x) = x^n$. Explain why the graph of $x = y^2$ is not the graph of a function of $x$. Is the graph of $x = y^3$ the graph of a function of $x$? If so, of what function of $x$ is it the graph? Determine for what integers $n$ the graph of $x = y^n$ is the graph of a function of $x$.

85 Discussion - Step Functions · Level 2

*Step Functions.* In Example 7 and Exercises 82 and 83 we are given functions whose graphs consist of horizontal line segments. Such functions are often called *step functions*, because their graphs look like stairs. Give some other examples of step functions that arise in everyday life.

86 Discussion - Stretched Step Functions · Level 3

*Stretched Step Functions.* Sketch graphs of the functions $f(x) = \lfloor x \rfloor$, $g(x) = \lfloor 2 x \rfloor$, and $h(x) = \lfloor 3 x \rfloor$ on separate graphs. How are the graphs related? If $n$ is a positive integer, what does the graph of $k(x) = \lfloor n x \rfloor$ look like?

87 Discussion - Graph of Absolute Value of a Function · Level 3

*Graph of the Absolute Value of a Function.* (a) Draw the graphs of the functions $f(x) = x^2 + x - 6$ and $g(x) = |x^2 + x - 6|$. How are the graphs of $f$ and $g$ related? (b) Draw the graphs of the functions $f(x) = x^4 - 6 x^2$ and $g(x) = |x^4 - 6 x^2|$. How are the graphs of $f$ and $g$ related? (c) In general, if $g(x) = |f(x)|$, how are the graphs of $f$ and $g$ related? Draw graphs to illustrate your answer.

88 Example - Graphing by Plotting Points · Level 1

Sketch graphs of the following functions:

(a) $f(x) = x^2$

Answer for 88(a)

(b) $g(x) = x^3$

Answer for 88(b)

(c) $h(x) = \sqrt{x}$

Answer for 88(c)

Enter your answer directly below each part above.

89 Example - Calculator Graphing · Level 2

Use a graphing calculator to graph the function $f(x) = x^3 - 8 x^2$ in an appropriate viewing rectangle.

90 Example - Family of Power Functions · Level 3

(a) Graph the functions $f(x) = x^n$ for $n = 2, 4$, and 6 in the viewing rectangle $[-2, 2]$ by $[-1, 3]$.

Answer for 90(a)

(b) Graph the functions $f(x) = x^n$ for $n = 1, 3$, and 5 in the viewing rectangle $[-2, 2]$ by $[-2, 2]$.

Answer for 90(b)

(c) What conclusions can you draw from these graphs?

Answer for 90(c)

Enter your answer directly below each part above.

91 Example - Piecewise Defined Function · Level 3

Sketch the graph of the piecewise defined function: $ f(x) = \begin{cases} x^2 & \quad \text{if } x \leq 1 \\ 2 x + 1 & \quad \text{if } x > 1 \end{cases} $

92 Example - Graph of Absolute Value Function · Level 2

Sketch a graph of the absolute value function $f(x) = |x|$.

93 Example - Graph of Greatest Integer Function · Level 2

Sketch a graph of $f(x) = \lfloor x \rfloor$, the greatest integer function.

94 Example - Step Function (Phone Call Cost) · Level 3

The cost of a long-distance daytime phone call from Toronto, Canada, to Mumbai, India, is 69 cents for the first minute and 58 cents for each additional minute (or part of a minute). Draw the graph of the cost $C$ (in dollars) of the phone call as a function of time $t$ (in minutes).

95 Example - Vertical Line Test · Level 2

Using the Vertical Line Test, determine which of the curves in parts (a), (b), (c), and (d) of Figure 11 represent functions.

96 Example - Equations That Define Functions · Level 2

Does the equation define $y$ as a function of $x$?

(a) $y - x^2 = 2$

Answer for 96(a)

(b) $x^2 + y^2 = 4$

Answer for 96(b)

Enter your answer directly below each part above.

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