Description

Question 1: (i) Determine if the following functions are even, odd, or neither. Show your algebraic justification foreach.(ii) Graph each functions using a graphing calculator or using an online graphing tool (such asDesmos.com). Provide a sketch or screenshot of your graphs and identify the symmetry for each.a) π(π₯) = The equation is given in the screenshotb) π(π₯) = The equation is given in the screenshotQuestion 2: Create.(a) Create your own piecewise-defined function to model a real-life situation such as your commute,a monthly salary (example #9, and problem #87 on page 212) or a cell phone plan (problem #88 page212). State the functionβs equation and give a description of the function stating the input and outputof your function.(b) Sketch an accurate graph of your newly-created function using appropriate axes values. (Notethat in some cases itβs appropriate to use non-integer values for the independent variable.) Includevalues where the function changes.(c) Evaluate the piecewise function at three values and interpret what these points represent in thecontext of your problem.(d) Identify the domain and range appropriate for your problem situation. Write your answers usingset-builder notation.(e) State where the function is increasing, decreasing or constant using interval notation. (section 1.7page 205)

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SECTION 1.7

Analyzing Graphs of Functions and Piecewise-Defined Functions

SECTION 1.7Analyzing Graphs of Functions and Piecewise-Defined Functions

OBJECTIVES

1. Test for Symmetry

2. Identify Even and Odd Functions

3. Graph Piecewise-Defined Functions

4. Investigate Increasing, Decreasing, and Constant Behavior of a Function

5. Determine Relative Minima and Maxima of a Function

1. Test for Symmetry

The photos in Figures 1-28 through 1-30 each show a type of symmetry.

Figure 1-28

Figure 1-29

Lecture: Introduction to Symmetry

PDF Transcript for Lecture: Introduction to Symmetry

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Figure 1-30

The photo of the kingfisher (Figure 1-28) shows an image of the bird reflected in the water. Suppose

that we superimpose the x-axis at the waterline. Every point (x, y) on the bird has a mirror image (x,

βy) below the x-axis. Therefore, this image is symmetric with respect to the x-axis.

A human face is symmetric with respect to a vertical line through the center (Figure 1-29). If we

place the y-axis along this line, a point (x, y) on one side has a mirror image at (βx, y). This image is

symmetric with respect to the y-axis.

The flower shown in Figure 1-30 is symmetric with respect to the point at its center. Suppose that we

place the origin at the center of the flower. Notice that a point (x, y) on the image has a

corresponding point (βx, βy) on the image. This image is symmetric with respect to the origin.

Given an equation in the variables x and y, use the following rules to determine if the graph is

symmetric with respect to the x-axis, the y-axis, or the origin.

Tests for Symmetry

Consider an equation in the variables x and y.

The graph of the equation is symmetric with respect to the y-axis if substituting βx for x in the

equation results in an equivalent equation.

The graph of the equation is symmetric with respect to the x-axis if substituting βy for y in the

equation results in an equivalent equation.

The graph of the equation is symmetric with respect to the origin if substituting βx for x and βy

for y in the equation results in an equivalent equation.

Lecture: Testing for Symmetry

PDF Transcript for Lecture: Testing for Symmetry

Page 198

EXAMPLE 1Testing for Symmetry

Determine whether the graph is symmetric with respect to the y-axis, x-axis, or origin.

a. y = |x|

b. x = y2 β 4

Solution:

a.

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The graph is symmetric with respect to the y-axis only.

b.

The graph is symmetric with respect to the x-axis only (Figure 1-31).

Figure 1-31

TIP

The graph of y = |x| is one of the basic graphs presented in Section 1.6. From our familiarity with the

graph we can visualize the symmetry with respect to the y-axis.

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Skill Practice 1

Determine whether the graph is symmetric with respect to the y-axis, x-axis, or origin.

a. y = x2

b. |y| = x + 1

Answers

1.

a. y-axis

b. x-axis

EXAMPLE 2Testing for Symmetry

Determine whether the graph is symmetric with respect to the y-axis, x-axis, or origin.

x2 + y2 = 9

Solution:

The graph of x2 + y2 = 9 is a circle with center at the origin and radius 3. By inspection, we can see

that the graph is symmetric with respect to both axes and the origin.

Page 199

The graph is symmetric with respect to the y-axis, the x-axis, and the origin.

Skill Practice 2

Determine whether the graph is symmetric with respect to the y-axis, x-axis, or origin.

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Answer

2. y-axis, x-axis, and origin

2. Identify Even and Odd Functions

A function may be symmetric with respect to the y-axis or to the origin. A function that is symmetric

with respect to the y-axis is called an even function. A function that is symmetric with respect to the

origin is called an odd function.

Even and Odd Functions

A function f is an even function if f(βx) = f(x) for all x in the domain of f. The graph of an even

function is symmetric with respect to the y-axis.

A function f is an odd function if f(βx) = βf(x) for all x in the domain of f. The graph of an odd

function is symmetric with respect to the origin.

Avoiding Mistakes

The only functions that are symmetric with respect to the x-axis are functions whose points lie solely

on the x-axis.

Lecture: Introduction to Even and Odd Functions

PDF Transcript for Lecture: Introduction to Even and Odd Functions

EXAMPLE 3Identifying Even and Odd Functions

By inspection determine if the function is even, odd, or neither.

Solution:

a.

The function is symmetric with respect to the origin. Therefore, the function is an odd function.

b.

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The function is symmetric with respect to the y-axis. Therefore, the function is an even

function.

c.

The function is not symmetric with respect to either the y-axis or the origin. Therefore, the

function is neither even nor odd.

Page 200

Skill Practice 3

Determine if the function is even, odd, or neither.

a.

b.

c.

Answers

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a. Even function

b. Odd function

c. Neither even nor odd

Lecture: Determining Whether a Function is Even Odd or Neither

PDF Transcript for Lecture: Determining Whether a Function is Even Odd or Neither

EXAMPLE 4Identifying Even and Odd Functions

Determine if the function is even, odd, or neither.

a. f(x) = β2×4 + 5|x|

b. g(x) = 4×3 β x

c. h(x) = 2×2 + x

Solution:

a.

b.

Since g(βx) = βg(x), the function g is an odd function.

c.

Since

, the function is not even.

Next, test whether h is an odd function. Test whether h(βx) = βh(x).

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Since

, the function is not an odd function. Therefore, h is neither even nor odd.

TIP

In Example 4(a), we suspect that f is an even function because each term is of the form xeven or |x|.

In each case, replacing x by βx results in an equivalent term.

TIP

In Example 4(b), we suspect that g is an odd function because each term is of the form xodd. In each

case, replacing x by βx results in the opposite of the original term.

TIP

In Example 4(c), h(x) has a mixture of terms of the form xodd and xeven. Therefore, we might

suspect that the function is neither even nor odd.

Skill Practice 4

Determine if the function is even, odd, or neither.

a. m(x) = βx5 + x3

b. n(x) = x2 β |x| + 1

c. p(x) = 2|x| + x

Answers

4.

a. Odd function

b. Even function

c. Neither even nor odd

Page 201

3. Graph Piecewise-Defined Functions

Suppose that a car is stopped for a red light. When the light turns green, the car undergoes a

constant acceleration for 20 sec until it reaches a speed of 45 mph. It travels 45 mph for 1 min (60

sec), and then decelerates for 30 sec to stop at another red light. The graph of the car’s speed y (in

mph) versus the time x (in sec) after leaving the first red light is shown in Figure 1-32.

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Figure 1-32

Notice that the graph can be segmented into three pieces. The first 20 sec is represented by a linear

function with a positive slope, y = 2.25x. The next 60 sec is represented by the constant function y =

45. And the last 30 sec is represented by a linear function with a negative slope, y = β1.5x + 165.

To write a rule defining this function we use a piecewise-defined function in which we define each

βpieceβ on a restricted domain.

Lecture: Interpreting a Piecewise-Defined Function

PDF Transcript for Lecture: Interpreting a Piecewise-Defined Function

EXAMPLE 5Interpreting a Piecewise-Defined Function

Evaluate the function for the given values of x.

a. f(β3)

b. f(β1)

c. f(2)

d. f(6)

Solution:

a.

b.

c.

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d.

Skill Practice 5

Evaluate the function for the given values of x.

a. f(β3)

b. f(β2)

c. f(1)

d. f(4)

Answers

5.

a. 4

b. 4

c. 3

d. 3

Graphing Calculator: Graphing a Piecewise-Defined Function on a Calculator

Page 202

TECHNOLOGY CONNECTIONS

Graphing a Piecewise-Defined Function

A graphing calculator can be used to graph a piecewise-defined function. The format to enter the

function is as follows.

Each condition in parentheses is an inequality and the calculator assigns it a value of 1 or 0

depending on whether the inequality is true or false. If an inequality is true, the function is divided by

1 on that interval and is βturned on.β If an inequality is false, then the function is divided by 0. Since

division by zero is undefined, the calculator does not graph the function on that interval, and the

function is effectively βturned off.β

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Enter the function from Example 5 as shown. Note that the inequality symbols can be found in the

TEST menu.

Notice that the individual βpiecesβ of the graph do not βhook-up.β For this reason, it is also a good

practice to put the calculator in DOT mode in the

menu.

In Examples 6 and 7, we graph piecewise-defined functions.

Animation: Graphing a piecewise-defined function

Lecture: Graphing a Piecewise-Defined Function

PDF Transcript for Lecture: Graphing a Piecewise-Defined Function

EXAMPLE 6Graphing a Piecewise-Defined Function

Graph the function defined by

.

Solution:

The first rule f(x) = β3x defines a line with slope β3 and y-intercept (0, 0). This line should be

graphed only to the left of x = 1. The point (1, β3) is graphed as an open dot, because the

point is not part of the rule f(x) = β3x. See the blue portion of the graph in Figure 1-33.

Figure 1-33

The second rule f(x) = β3 is a horizontal line for x β₯ 1. The point (1, β3) is a closed dot to show

that it is part of the rule f(x) = β3. The closed dot from the red segment of the graph βoverridesβ

the open dot from the blue segment. Taken together, the closed dot βplugsβ the hole in the

graph.

Page 203

Skill Practice 6

Graph the function.

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Answer

6.

TIP

The function in Example 6 has no βgaps,β and therefore we say that the function is continuous.

Informally, this means that we can draw the function without lifting our pencil from the page. The

formal definition of a continuous function will be studied in calculus.

EXAMPLE 7Graphing a Piecewise-Defined Function

Graph the function.

Solution:

The first rule f(x) = x + 3 defines a line with slope 1 and y-intercept (0, 3). This line should be

graphed only for x < β1 (that is to the left of x = β1). The point (β1, 2) is graphed as an open dot,
because the point is not part of the function. See the red portion of the graph in Figure 1-34.
Figure 1-34
The second rule f(x) = x2 is one of the basic functions learned in Section 1.6. It is a parabola with
vertex at the origin. We sketch this function only for x values on the interval β1 β€ x < 2. The point
(β1, 1) is a closed dot to show that it is part of the function. The point (2, 4) is displayed as an open
dot to indicate that it is not part of the function.
Avoiding Mistakes
Note that the function cannot have a closed dot at both (β1, 1) and (β1, 2) because it would not pass
the vertical line test.
TIP
The function in Example 7 has a gap at x = β1, and therefore, we say that f is discontinuous at β1.
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Skill Practice 7
Graph the function.
Answer
7.
We now look at a special category of piecewise-defined functions called step functions. The graph
of a step function is a series of discontinuous βsteps.β One important step function is called the
greatest integer function or floor function. It is defined by
where
is the greatest integer less than or equal to x.
The operation
may also be denoted as int(x) or by floor(x). These alternative notations are often
used in computer programming.
In Example 8, we graph the greatest integer function.
Lecture: Introduction to the Greatest Integer Function
PDF Transcript for Lecture: Introduction to the Greatest Integer Function
Page 204
EXAMPLE 8Graphing the Greatest Integer Function
Graph the function defined by
.
Solution:
x
Evaluate f for several values of x.
β1.7
β2
Greatest integer less than or equal to β1.7 is β2.
β1
β1
Greatest integer less than or equal to β1 is β1.
β0.6
β1
Greatest integer less than or equal to β0.6 is β1.
0
0
Greatest integer less than or equal to 0 is 0.
0.4
0
Greatest integer less than or equal to 0.4 is 0.
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1
1
Greatest integer less than or equal to 1 is 1.
1.8
1
Greatest integer less than or equal to 1.8 is 1.
2
2
Greatest integer less than or equal to 2 is 2.
2.5
2
Greatest integer less than or equal to 2.5 is 2.
TIP
On many graphing calculators, the greatest integer function is denoted by int( ) and is found under
the MATH menu followed by NUM.
From the table, we see a pattern and from the pattern, we form the graph.
Skill Practice 8
Evaluate f(x) =
for the given values of x.
a. f(1.7)
b. f(5.5)
c. f(β4)
d. f(β4.2)
Answers
8.
a. 1
b. 5
c. β4
d. β5
In Example 9, we use a piecewise-defined function to model an application.
EXAMPLE 9Using a Piecewise-Defined Function in an Application
A salesperson makes a monthly salary of $3000 along with a 5% commission on sales over $20,000
for the month. Write a piecewise-defined function to represent the salesperson's monthly income I(x)
(in $) for x dollars in sales.
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Solution:
Let x represent the amount in sales.
Then x β 20,000 represents the amount in sales over $20,000.
There are two scenarios for the salesperson's income.
Scenario The salesperson sells $20,000 or less. In this case, the monthly income is a constant
1:
$3000. This is represented by
Scenario The salesperson sells over $20,000. In this case, the monthly income is $3000 plus 5% of
2:
sales over $20,000. This is represented by
Page 205
Therefore, a piecewise-defined function for monthly income is
Alternatively, we can simplify to get
A graph of y = I(x) is shown in Figure 1-35. Notice that for x = $20,000, both equations within the
piecewise-defined function yield a monthly salary of $3000. Therefore, the two line segments in the
graph meet at (20,000, 3000).
Figure 1-35
Skill Practice 9
A retail store buys T-shirts from the manufacturer. The cost is $7.99 per shirt for 1 to 100 shirts,
inclusive. Then the price is decreased to $6.99 per shirt thereafter. Write a piecewise-defined
function that expresses the cost C(x) (in $) to buy x shirts.
Answer
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9.
4. Investigate Increasing, Decreasing, and Constant Behavior of a Function
The graph in Figure 1-36 approximates the altitude of an airplane, f(t), at a time t minutes after
takeoff.
Figure 1-36
Notice that the plane's altitude increases up to the first 40 min of the flight. So we say that the
function f is increasing on the interval (0, 40). The plane flies at a constant altitude for the next 1 hr
20 min, so we say that f is constant on the interval (40, 120). Finally, the plane's altitude decreases
for the last 40 min, so we say that f is decreasing on the interval (120, 160).
Informally, a function is increasing on an interval in its domain if its graph rises from left to right. A
function is decreasing on an interval in its domain if the graph βfallsβ from left to right. A function is
constant on an interval in its domain if its graph is horizontal over the interval. These ideas are
stated formally using mathematical notation.
Animation: Identifying the intervals of increasing decreasing and constant behavior on closed
intervals
Animation: Identifying the intervals of increasing decreasing and constant behavior on open intervals
Lecture: Introduction to Increasing and Decreasing Functions
PDF Transcript for Lecture: Introduction to Increasing and Decreasing Functions
Page 206
Intervals of Increasing, Decreasing, and Constant Behavior
Suppose that I is an interval contained within the domain of a function f.
f is increasing on I if f(x1) < f(x2) for all x1 < x2 on I.
f is decreasing on I if f(x1) > f(x2) for all x1 < x2 on I.
f is constant on I if f(x1) = f(x2) for all x1 and x2 on I.
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Lecture: Determining the Intervals over Which a Function Increases Decreases or is Constant with
Closed Intervals
Lecture: Determining the Intervals over Which a Function Increases Decreases or is Constant with
Open Intervals
PDF Transcript for Lecture: Determining the Intervals over Which a Function Increases Decreases
or is Constant with Closed Intervals
PDF Transcript for Lecture: Determining the Intervals over Which a Function Increases Decreases
or is Constant with Open Intervals
EXAMPLE 10Determining the Intervals Over Which a Function Is Increasing, Decreasing, and
Constant
Use interval notation to write the interval(s) over which f is
a. Increasing
b. Decreasing
c. Constant
Solution:
a. f is increasing on the interval (2, 3). (Highlighted in red tint.)
b. f is decreasing on the interval (β4, β1) and (3, β). (Highlighted in orange tint.)
c. f is constant on the interval (β1, 2). (Highlighted in green tint.)
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Skill Practice 10
Use interval notation to write the interval(s) over which f is
a. Increasing
b. Decreasing
c. Constant
Answers
10.
a. (β5, β3)
(β1, 2)
b. (2, β)
c. (β3, β1)
Page 207
5. Determine Relative Minima and Maxima of a Function
The intervals over which a function changes from increasing to decreasing behavior or vice versa
tell us where to look for relative maximum values and relative minimum values of a function.
Consider the function pictured in Figure 1-37.
Figure 1-37
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The function has a relative maximum of f(a). Informally, this means that f(a) is the greatest
function value relative to other points on the function nearby.
The function has a relative minimum of f(b). Informally, this means that f(b) is the smallest
function value relative to other points on the function nearby.
This is stated formally in the following definition.
Relative Minimum and Relative Maximum Values
f(a) is a relative maximum of f if there exists an open interval containing a such that f(a) β₯ f(x)
for all x in the interval.
f(b) is a relative minimum of f if there exists an open interval containing b such that f(b) β€ f(x)
for all x in the interval.
Note: An open interval is an interval in which the endpoints are not included.
TIP
The plural of maximum and minimum are maxima and minima.
Note that relative maxima and minima are also called local maxima and minima.
If an ordered pair (a, f(a)) corresponds to a relative minimum or relative maximum, we interpret the
coordinates of the ordered pair as follows.
The x-coordinate is the location of the relative maximum or minimum within the domain of the
function.
The y-coordinate is the value of the relative maximum or minimum. This tells us how βhighβ or
βlowβ the graph is at that point.
Lecture: Introduction to Relative Maxima and Minima
PDF Transcript for Lecture: Introduction to Relative Maxima and Minima
EXAMPLE 11Finding Relative Maxima and Minima
For the graph of y = g(x) shown,
a. Determine the location and value of any relative maxima.
b. Determine the location and value of any relative minima.
Page 208
Solution:
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a. The point (2, 1) is the highest point in a small interval surrounding x = 2. Therefore, at x = 2,
the function has a relative maximum of 1.
b. The point (β1, β5) is the lowest point in a small interval surrounding x = β1. Therefore, at x =
β1, the function has a relative minimum of β5.
The point (4, β2) is the lowest point in a small interval surrounding x = 4. Therefore, at x = 4,
the function has a relative minimum of β2.
Avoiding Mistakes
Be sure to note that the value of a relative minimum or relative maximum is the y value of a function,
not the x value.
Skill Practice 11
For the graph shown,
a. Determine the location and value of any relative maxima.
b. Determine the location and value of any relative minima.
Answers
11.
a. At x = β2, the function has a relative maximum of 3.
b. At x = 2, the function has a relative minimum of 0.
Graphing Calculator: Using a Graphing Calculator to Find the relative Minimum or
Maximum of a Function
TECHNOLOGY CONNECTIONS
Determining Relative Maxima and Minima
Relative maxima and relative minima are often difficult to find analytically and require techniques
from calculus. However, a graphing utility can be used to approximate the location and value of
relative maxima and minima. To do so, we use the Minimum and Maximum features.
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For example, enter the function defined by Y1 = x3 β 4x2 + 3x. Then access the Maximum feature
from the CALC menu.
The calculator asks for a left bound. This is a point slightly to the left of the relative maximum. Then
hit ENTER.
The calculator asks for a right bound. This is a point slightly to the right of the relative maximum. Hit
ENTER.
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The calculator asks for a guess. This is a point close to the relative maximum. Hit ENTER and the
approximate coordinates of the relative maximum point are shown (0.45, 0.63).
To find the relative minimum, repeat these steps using the Minimum feature. The coordinates of the
relative minimum point are approximately (2.22, β2.11).
SECTION 1.7Practice Exercises
Prerequisite Review
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R.1. Given the function defined by
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, find f(βa).
Answers
For Exercises R.2βR.4, graph the set and express the set in interval notation.
R.2.
Answers
R.3.
Answers
R.4.
Answers
Concept Connections
1. A graph of an equation is symmetric with respect to the
results in an equivalent equation.
-axis if replacing x by βx
Answers
2. A graph of an equation is symmetric with respect to the
results in an equivalent equation.
-axis if replacing y by βy
3. A graph of an equation is symmetric with respect to the
y by βy results in an equivalent equation.
if replacing x by βx and
Answers
4. An even function is symmetric with respect to the
5. An odd function is symmetric with respect to the
.
.
Answers
6. The expression
represents the greatest integer, less than or equal to x.
Objective 1: Test for Symmetry
For Exercises 7β18, determine whether the graph of the equation is symmetric with respect
to the x-axis, y-axis, origin, or none of these. (See Examples 1β2)
7. y = x2 + 3
Answers
8. y = β|x| β 4
9. x = β|y| β 4
Answers
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10. x = y2 + 3
Exercise: Testing for Symmetry
PDF Transcript for Exercise: Testing for Symmetry
11. x2 + y2 = 3
Answers
12. |x| + |y| = 4
Exercise: Testing for Symmetry
PDF Transcript for Exercise: Testing for Symmetry
13. y = |x| + 2x + 7
Answers
14. y = x2 + 6x + 1
15. x2 = 5 + y2
Answers
16. y4 = 2 + x2
17.
Answers
18.
Objective 2: Identify Even and Odd Functions
19. What type of symmetry does an even function have?
Answers
20. What type of symmetry does an odd function have?
Page 210
For Exercises 21β26, use the graph to determine if the function is even, odd, or neither. (See
Example 3)
21.
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22.
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Answers
23.
24.
Answers
25.
26.
Answers
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27.
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a. Given f(x) = 4x2 β 3|x|, find f(βx).
b. Is f(βx) = f(x)?
c. Is this function even, odd, or neither?
Answers
28.
a. Given g(x) = βx8 + |3x|, find g(βx).
b. Is g(βx) = g(x)?
c. Is this function even, odd, or neither?
29.
a. Given h(x) = 4x3 β 2x, find h(βx).
b. Find βh(x).
c. Is h(βx) = βh(x)?
d. Is this function even, odd, or neither?
Answers
30.
a. Given k(x) = β8x5 β 6x3, find k(βx).
b. Find βk(x).
c. Is k(βx) = βk(x)?
d. Is this function even, odd, or neither?
31.
a. Given m(x) = 4x2 + 2x β 3, find m(βx).
b. Find βm(x).
c. Is m(βx) = m(x)?
d. Is m(βx) = βm(x)?
e. Is this function even, odd, or neither?
Answers
32.
a. Given n(x) = 7|x| + 3x β 1, find n(βx).
b. Find βn(x).
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c. Is n(βx) = n(x)?
d. Is n(βx) = βn(x)?
e. Is this function even, odd, or neither?
For Exercises 33β46, determine if the function is even, odd, or neither. (See Example 4)
33.
Answers
34.
Exercise: Determining Whether a Function is Even Odd or Neither
PDF Transcript for Exercise: Determining Whether a Function is Even Odd or Neither
35.
Answers
36.
37.
Answers
38.
39.
Exercise: Determining Whether a Function is Even Odd or Neither
PDF Transcript for Exercise: Determining Whether a Function is Even Odd or Neither
Answers
40.
41.
Answers
42.
43.
Answers
44.
45.
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Answers
46.
Objective 3: Graph Piecewise-Defined Functions
For Exercises 47β50, evaluate the function for the given values of x. (See Example 5)
47.
a. f(3)
b. f(β2)
c. f(β1)
d. f(4)
e. f(5)
Exercise: Interpreting a Piecewise-Defined Function
PDF Transcript for Exercise: Interpreting a Piecewise-Defined Function
Answers
48.
a. g(β3)
b. g(3)
c. g(β2)
d. g(0)
e. g(4)
Page 211
49.
a. h(β1.7)
b. h(β2.5)
c. h(0.05)
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d. h(β2)
e. h(0)
Answers
50.
a. t(1.99)
b. t(0.4)
c. t(3)
d. t(1)
e. t(3.001)
51. A sled accelerates down a hill and then slows down after it reaches a flat portion of ground.
The speed of the sled s(t) (in ft/sec) at a time t (in sec) after movement begins can be
approximated by:
Determine the speed of the sled after 10 sec, 20 sec, 30 sec, and 40 sec. Round to 1 decimal
place if necessary.
Answers
52. A car starts from rest and accelerates to a speed of 60 mph in 12 sec. It travels 60 mph for 1
min and then decelerates for 20 sec until it comes to rest. The speed of the car s(t) (in mph) at
a time t (in sec) after the car begins motion can be modeled by:
Determine the speed of the car 6 sec, 12 sec, 45 sec, and 80 sec after the car begins motion.
For Exercises 53β56, match the function with its graph.
53. f(x) = x + 1 for x < 2
Answers
54. f(x) = x + 1 for β1 < x β€ 2
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55. f(x) = x + 1 for β1 β€ x < 2
Answers
56. f(x) = x + 1 for x β₯ 2
a.
b.
c.
d.
57.
a. Graph p(x) = x + 2 for x β€ 0. (See Examples 6β7)
b. Graph q(x) = βx2 for x > 0.

c.

Answers

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58.

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a. Graph f(x) = |x| for x < 0.
b.
c.
59.
a.
b.
c.
Answers
60.
a. Graph a(x) = x for x < 1.
b.
c.
For Exercises 61β70, graph the function. (See Examples 6β7)
61.
Answers
62.
63.
Answers
64.
65.
Answers
66.
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67.
Answers
68.
Exercise: Graphing a Piecewise-Defined Function
PDF Transcript for Exercise: Graphing a Piecewise-Defined Function
69.
Answers
70.
71.
a.
b. To what basic function from Section 1.6 is the graph of f equivalent?
Answers
For Exercises 72β80, evaluate the step function defined by
(See Example 8)
for the given values of x.
72. f(β3.7)
73. f(β4.2)
Answers
74. f(β0.5)
75. f(β0.09)
Answers
76. f(0.5)
77. f(0.09)
Answers
78. f(6)
79. f(β9)
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Answers
80. f(β5)
For Exercises 81β84, graph the function. (See Example 8)
81.
Answers
82.
83.
Answers
84. h(x) = int(2x)
85. For a recent year, the rate for first class postage was as follows. (See Example 9)
Weight not OverPrice
1 oz
$0.44
2 oz
$0.61
3 oz
$0.78
3.5 oz
$0.95
Write a piecewise-defined function to model the cost C(x) to mail a letter first class if the letter
is x ounces.
Answers
86. The water level in a retention pond started at 5 ft (60 in.) and decreased at a rate of 2 in./day
during a 14-day drought. A tropical depression moved through at the beginning of the 15th day
and produced rain at an average rate of 2.5 in./day for 5 days. Write a piecewise-defined
function to model the water level L(x) (in inches) as a function of the number of days x since
the beginning of the drought.
87. A salesperson makes a base salary of $2000 per month. Once he reaches $40,000 in total
sales, he earns an additional 5% commission on the amount in sales over $40,000. Write a
piecewise-defined function to model the salesperson's total monthly salary S(x) (in $) as a
function of the amount in sales x.
Answers
88. A cell phone plan charges $49.95 per month plus $14.02 in taxes, plus $0.40 per minute for
calls beyond the 600-min monthly limit. Write a piecewise-defined function to model the
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monthly cost C(x) (in $) as a function of the number of minutes used x for the month.
Objective 4: Investigate Increasing, Decreasing, and Constant Behavior of a Function
For Exercises 89β96, use interval notation to write the intervals over which f is (a) increasing,
(b) decreasing, and (c) constant. (See Example 10)
89.
Exercise: Determining the Intervals over Which a Function Increases Decreases or Constant
with Closed Intervals
Exercise: Determining the Intervals over Which a Function Increases Decreases or Constant
with Open Intervals
PDF Transcript for Exercise: Determining the Intervals over Which a Function Increases
Decreases or Constant with Closed Intervals
PDF Transcript for Exercise: Determining the Intervals over Which a Function Increases
Decreases or Constant with Open Intervals
90.
Answers
91.
92.
Answers
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93.
94.
95.
96.
Answers
Page 213
Answers
Objective 5: Determine Relative Minima and Maxima of a Function
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For Exercises 97β102, identify the location and value of any relative maxima or minima of the
function. (See Example 11)
97.
98.
Answers
Exercise: Introduction to Relative Maxima and Minima
PDF Transcript for Exercise: Introduction to Relative Maxima and Minima
99.
100.
Answers
101.
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102.
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Answers
103. The graph shows the depth d (in ft) of a retention pond, t days after recording began.
a. Over what interval(s) does the depth increase?
b. Over what interval(s) does the depth decrease?
c. Estimate the times and values of any relative maxima or minima on the interval (0, 20).
d. If rain is the only water that enters the pond, explain what the intervals of increasing and
decreasing behavior mean in the context of this problem.
Answers
104. The graph shows the height h (in meters) of a roller coaster t seconds after the ride starts.
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a. Over what interval(s) does the height increase?
b. Over what interval(s) does the height decrease?
c. Estimate the times and values of any relative maxima or minima on the interval (0, 70).
Page 214
Mixed Exercises
For Exercises 105β110, produce a rule for the function whose graph is shown. (Hint:
Consider using the basic functions learned in Section 1.6 and transformations of their
graphs.)
105.
106.
Answers
107.
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108.
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Answers
109.
110.
Answers
For Exercises 111β112,
a. Graph the function.
b. Write the domain in interval notation.
c. Write the range in interval notation.
d. Evaluate f(β1), f(1), and f(2).
e. Find the value(s) of x for which f(x) = 6.
f. Find the value(s) of x for which f(x) = β3.
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g. Use interval notation to write the intervals over which f is increasing, decreasing, or constant.
111.
Answers
112.
In computer programming, the greatest integer function is sometimes called the βfloorβ
function. Programmers also make use of the βceilingβ function, which returns the smallest
integer not less than x. For example: ceil(3.1) = 4. For Exercises 113β114, evaluate the floor
and ceiling functions for the given value of x.
floor(x) is the greatest integer less than or equal to x.
ceil(x) is the smallest integer not less than x.
113.
a. floor(2.8)
b. floor(β3.1)
c. floor(4)
d. ceil(2.8)
e. ceil(β3.1)
f. ceil(4)
Answers
114.
a. floor(5.5)
b. floor(β0.1)
c. floor(β2)
d. ceil(5.5)
e. ceil(β0.1)
f. ceil(β2)
Write About It
115. From an equation in x and y, explain how to determine whether the graph of the equation is
symmetric with respect to the x-axis, y-axis, or origin.
Answers
116. From the graph of a function, how can you determine if the function is even or odd?
117. Explain why the relation defined by
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is not a function.
Answers
118. Explain why the function is discontinuous at x = 1.
Page 215
119. Provide an informal explanation of a relative maximum.
Answers
120. Explain what it means for a function to be increasing on an interval.
Expanding Your Skills
121. Suppose that the average rate of change of a continuous function between any two points to
the left of x = a is negative, and the average rate of change of the function between any two
points to the right of x = a is positive. Does the function have a relative minimum or maximum
at a?
Answers
122. Suppose that the average rate of change of a continuous function between any two points to
the left of x = a is positive, and the average rate of change of the function between any two
points to the right of x = a is negative. Does the function have a relative minimum or maximum
at a?
A graph is concave up on a given interval if it βbendsβ upward. A graph is concave down on
a given interval if it βbendsβ downward. For Exercises 123β126, determine whether the curve
is (a) concave up or concave down and (b) increasing or decreasing.
123.
124.
Answers
125.
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126.
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Answers
127. For a recent year, the federal income tax owed by a taxpayer (singleβno dependents) was
based on the individual's taxable income. (Source: Internal Revenue Service, www.irs.gov)
If your taxable income is
overβ but not overβThe tax is
of the amount overβ
$0
$0
$8925
$0 + 10%
$8925 $36,250
$892.50 + 15% $8925
$36,250$87,850
$4991.25 + 25%$36,250
Write a piecewise-defined function that expresses an individual's federal income tax f(x) (in $)
as a function of the individual's taxable income x (in $).
Answers
Technology Connections
For Exercises 128β131, use a graphing utility to graph the piecewise-defined function.
128.
129.
Exercise: Graphing a Piecewise-Defined Function on a Calculator
Answers
130.
131.
For Exercises 132β135, use a graphing utility to
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a. Find the locations and values of the relative maxima and relative minima of the function on the
standard viewing window. Round to 3 decimal places.
b. Use interval notation to write the intervals over which f is increasing or decreasing.
Answers
132.
133.
Answers
134.
135.
Answers
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