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SECTION P.3 Functions and Their Graphs 19

Section P.3 Functions and Their Graphs

• Use function notation to represent and evaluate a function.

• Find the domain and range of a function.

• Sketch the graph of a function.

• Identify different types of transformations of functions.

• Classify functions and recognize combinations of functions.

Functions and Function Notation

A relation between two sets and is a set of ordered pairs, each of the form

where is a member of and is a member of A function from to is a relation

between and that has the property that any two ordered pairs with the same

-value also have the same -value. The variable is the independent variable, and

the variable is the dependent variable.

Many real-life situations can be modeled by functions. For instance, the area of

a circle is a function of the circle’s radius

is a function of

In this case is the independent variable and is the dependent variable.

Functions can be specified in a variety of ways. In this text, however, we will con-

centrate primarily on functions that are given by equations involving the dependent

and independent variables. For instance, the equation

Equation in implicit form

defines the dependent variable, as a function of the independent variable. To

evaluate this function (that is, to find the -value that corresponds to a given -value),

it is convenient to isolate on the left side of the equation.

Equation in explicit form

Using as the name of the function, you can write this equation as

Function notation

The original equation, implicitly defines as a function of When you

solve the equation for you are writing the equation in explicit form.

Function notation has the advantage of clearly identifying the dependent variable

as while at the same time telling you that is the independent variable and that

the function itself is “ ” The symbol is read “ of ” Function notation allows

you to be less wordy. Instead of asking “What is the value of that corresponds to

” you can ask “What is ”fs3d?x 5 3?

y

x.ffsxdf.

xfsxd

y,

x.yx2 1 2y 5 1,

fsxd 51

2s1 2 x2d.

f

y 51

2s1 2 x2d

y

xy

x,y,

x2 1 2y 5 1

Ar

r.AA 5 pr2

r.

A

y

xyx

YX

YXY.yXx

sx, yd,YX

Range

x

f

Domain

y = f (x)

Y

X

A real-valued function of a real variable

Figure P.22

f

FUNCTION NOTATION

The word function was first used by Gottfried

Wilhelm Leibniz in 1694 as a term to denote

any quantity connected with a curve, such as

the coordinates of a point on a curve or the

slope of a curve. Forty years later, Leonhard

Euler used the word “function”to describe any

expression made up of a variable and some

constants. He introduced the notation

y 5 f sxd.

Definition of a Real-Valued Function of a Real Variable

Let and be sets of real numbers. A real-valued function of a real variable

from to is a correspondence that assigns to each number in exactly one

number in

The domain of is the set The number is the image of under and is

denoted by which is called the value of at The range of is a subset

of and consists of all images of numbers in (see Figure P.22).XY

fx.ff sxd,fxyX.f

Y.y

XxYXx

fYX

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20 CHAPTER P Preparation for Calculus

In an equation that defines a function, the role of the variable is simply that of

a placeholder. For instance, the function given by

can be described by the form

where parentheses are used instead of To evaluate simply place in each

set of parentheses.

Substitute for

Simplify.

Simplify.

NOTE Although is often used as a convenient function name and as the independent

variable, you can use other symbols. For instance, the following equations all define the same

function.

Function name is independent variable is



EXAMPLE 1 Evaluating a Function

For the function defined by evaluate each expression.

a. b. c.

Solution

a. Substitute for

Simplify.

b. Substitute for

Expand binomial.

Simplify.

c.

NOTE The expression in Example 1(c) is called a difference quotient and has a special

significance in calculus. You will learn more about this in Chapter 2.

Dx Þ 0 5 2x 1 Dx,

5Dxs2x 1 Dxd

Dx

52xDx 1 sDxd2

Dx

5x2 1 2xDx 1 sDxd2 1 7 2 x2 2 7

Dx

f sx 1 Dxd 2 f sxd

Dx5

fsx 1 Dxd2 1 7g 2 sx2 1 7dDx

5 b2 2 2b 1 8

5 b2 2 2b 1 1 1 7

x.b 2 1 f sb 2 1d 5 sb 2 1d2 1 7

5 9a2 1 7

x.3af s3ad 5 s3ad2 1 7

Dx Þ 0f sx 1 Dxd 2 f sxd

Dx,f sb 2 1df s3ad

f sxd 5 x2 1 7,f

s.g, gssd 5 s2 2 4s 1 7

t.f, fstd 5 t2 2 4t 1 7

x.f, fsxd 5 x2 2 4x 1 7

xf

5 17

5 2s4d 1 8 1 1

x.22 fs22d 5 2s22d2 2 4s22d 1 1

22f s22d,x.

f sjd 5 2sjd22 4sjd 1 1

fsxd 5 2x2 2 4x 1 1

x

STUDY TIP In calculus, it is important

to communicate clearly the domain of a

function or expression. For instance, in

Example 1(c) the two expressions

are equivalent because is exclud-

ed from the domain of each expression.

Without a stated domain restriction, the

two expressions would not be equivalent.

Dx 5 0

Dx Þ 0

f sx 1 Dxd 2 f sxdD x

and 2x 1 Dx,


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The Domain and Range of a Function

The domain of a function can be described explicitly, or it may be described implicitly

by an equation used to define the function. The implied domain is the set of all real

numbers for which the equation is defined, whereas an explicitly defined domain is

one that is given along with the function. For example, the function given by

has an explicitly defined domain given by On the other hand, the

function given by

has an implied domain that is the set

EXAMPLE 2 Finding the Domain and Range of a Function

a. The domain of the function

is the set of all -values for which which is the interval To find

the range observe that is never negative. So, the range is the

interval as indicated in Figure P.23(a).

b. The domain of the tangent function, as shown in Figure P.23(b),

is the set of all -values such that

is an integer. Domain of tangent function

The range of this function is the set of all real numbers. For a review of the

characteristics of this and other trigonometric functions, see Appendix D.

EXAMPLE 3 A Function Defined by More than One Equation

Determine the domain and range of the function.

Solution Because is defined for and the domain is the entire set of

real numbers. On the portion of the domain for which the function behaves as

in Example 2(a). For the values of are positive. So, the range of the

function is the interval . (See Figure P.24.)

A function from to is one-to-one if to each -value in the range there

corresponds exactly one -value in the domain. For instance, the function given in

Example 2(a) is one-to-one, whereas the functions given in Examples 2(b) and 3 are

not one-to-one. A function from to is onto if its range consists of all of Y.YX

x

yYX

f0, `d1 2 xx < 1,

x ≥ 1,

x ≥ 1,x < 1f

fsxd 5 51 2 x,

!x 2 1,

if x < 1

if x ≥ 1

nx Þp

21 np,

x

fsxd 5 tan x

f0, `d,f sxd 5 !x 2 1

f1, `d.x 2 1 ≥ 0,x

fsxd 5 !x 2 1

Hx: x Þ ±2J.

gsxd 51

x2 2 4

Hx: 4 ≤ x ≤ 5J.

4 ≤ x ≤ 5fsxd 51

x2 2 4,

43

2

1

21x

Domain: x ≥ 1

Ran

ge:

y ≥

0 f (x) = x − 1

y

(a) The domain of is and the range is

f0, `d.f1, `df

2ππ

y

x

3

2

1

Ran

ge

Domain

f (x) = tan x

(b) The domain of is all -values such that

and the range is

Figure P.23

s2`, `d.x Þp

21 np

xf

43

2

1

21x

y

Ran

ge:

y ≥

0

Domain: all real x

x − 1, x ≥ 1f (x) =

1 − x, x < 1

The domain of is and the range

is

Figure P.24

f0, `d.

s2`, `df


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The Graph of a Function

The graph of the function consists of all points where is in the

domain of In Figure P.25, note that

the directed distance from the -axis

the directed distance from the -axis.

A vertical line can intersect the graph of a function of at most once. This

observation provides a convenient visual test, called the Vertical Line Test, for

functions of That is, a graph in the coordinate plane is the graph of a function of

if and only if no vertical line intersects the graph at more than one point. For exam-

ple, in Figure P.26(a), you can see that the graph does not define as a function of

because a vertical line intersects the graph twice, whereas in Figures P.26(b) and (c),

the graphs do define as a function of

Figure P.27 shows the graphs of eight basic functions. You should be able to

recognize these graphs. (Graphs of the other four basic trigonometric functions are

shown in Appendix D.)

x.y

xy

fx.

x

x fsxd 5

y x 5

f.

xsx, f sxdd,y 5 f sxd


x

−2 −1 1 2

2

1

−1

−2

f (x) = xy

Identity function

x

−2 −1 1 2

2

3

4

1

f (x) = x2y

Squaring function

x

−2 −1 1 2

2

1

−1

−2

f (x) = x3

y

Cubing function

x

1 2 3 4

2

3

4

1

f (x) = x

y

Square root function

x

−2 −1 1 2

2

3

4

1

x f (x) = x

y

Absolute value function

The graphs of eight basic functions

Figure P.27

x

−1 1 2

2

1

−1

f (x) =

y

1x

Rational function

x

2

1

−2

ππ π2−

f (x) = sin x

y

Sine function

x

2

1

−2

−1

ππ π2π−2 −

f (x) = cos x

y

Cosine function

x

x

f (x)

(x, f (x))y = f (x)y

The graph of a function

Figure P.25

x

−3 −2 1

4

2

y

(a) Not a function of

Figure P.26

x

x

3

2

1

−2

1 2 4

y

(b) A function of x

x

−1 1 2 3

4

1

3

y

(c) A function of x


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Transformations of Functions

Some families of graphs have the same basic shape. For example, compare the graph

of with the graphs of the four other quadratic functions shown in Figure P.28.

Each of the graphs in Figure P.28 is a transformation of the graph of The

three basic types of transformations illustrated by these graphs are vertical shifts,

horizontal shifts, and reflections. Function notation lends itself well to describing

transformations of graphs in the plane. For instance, if is considered to be

the original function in Figure P.28, the transformations shown can be represented by

the following equations.

Vertical shift up 2 units

Horizontal shift to the left 2 units

Reflection about the -axis

Shift left 3 units, reflect about -axis, and shift up 1 unitxy 5 2f sx 1 3d 1 1

xy 5 2f sxdy 5 f sx 1 2dy 5 f sxd 1 2

fsxd 5 x2

y 5 x2.

y 5 x2

x

−2 −1 1 2

3

4

1

y = x2 + 2

y = x2

y

(a) Vertical shift upward

x

−2−3 −1 1

3

4

1y = x2

y = (x + 2)2

y

(b) Horizontal shift to the left

x

−2 −1 1 2

1

2

y = x2

y = −x2

y

−1

−2

(c) Reflection

Figure P.28

x

−5 −3 −1 1 2

−2

1

2

3

4

y = 1 − (x + 3)2

y = x2

y

(d) Shift left, reflect, and shift upward

Basic Types of Transformations

Original graph:

Horizontal shift units to the right:

Horizontal shift units to the left:

Vertical shift units downward:

Vertical shift units upward:

Reflection (about the -axis):

Reflection (about the -axis):

Reflection (about the origin): y 5 2f s2xdy 5 f s2xdy

y 5 2fsxdx

y 5 fsxd 1 cc

y 5 fsxd 2 cc

y 5 fsx 1 cdc

y 5 fsx 2 cdc

y 5 fsxd

sc > 0d

E X P L O R A T I O N

Writing Equations for Functions

Each of the graphing utility screens

below shows the graph of one of the

eight basic functions shown on page

22. Each screen also shows a trans-

formation of the graph. Describe the

transformation. Then use your

description to write an equation for

the transformation.

a.

b.

c.

d.

−6 6

−3

5

−8 10

−4

8

8 108 10

−6 6

−4

4

6 66 6

−9 9

−3

9

9 99 9


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Classifications and Combinations of Functions

The modern notion of a function is derived from the efforts of many seventeenth- and

eighteenth-century mathematicians. Of particular note was Leonhard Euler, to whom

we are indebted for the function notation By the end of the eighteenth

century, mathematicians and scientists had concluded that many real-world phenomena

could be represented by mathematical models taken from a collection of functions

called elementary functions. Elementary functions fall into three categories.

1. Algebraic functions (polynomial, radical, rational)

2. Trigonometric functions (sine, cosine, tangent, and so on)

3. Exponential and logarithmic functions

You can review the trigonometric functions in Appendix D. The other nonalgebraic

functions, such as the inverse trigonometric functions and the exponential and

logarithmic functions, are introduced in Chapter 5.

The most common type of algebraic function is a polynomial function

where the positive integer is the degree of the polynomial function. The constants

are coefficients, with the leading coefficient and the constant term of the

polynomial function. It is common practice to use subscript notation for coefficients

of general polynomial functions, but for polynomial functions of low degree, the

following simpler forms are often used.

Zeroth degree: Constant function

First degree: Linear function

Second degree: Quadratic function

Third degree: Cubic function

Although the graph of a nonconstant polynomial function can have several turns,

eventually the graph will rise or fall without bound as moves to the right or left.

Whether the graph of

eventually rises or falls can be determined by the function’s degree (odd or even) and

by the leading coefficient as indicated in Figure P.29. Note that the dashed portions

of the graphs indicate that the Leading Coefficient Test determines only the right and

left behavior of the graph.

an,

fsxd 5 anxn 1 an21xn21 1 . . . 1 a2x2 1 a1x 1 a0

x

fsxd 5 ax3 1 bx2 1 cx 1 d

fsxd 5 ax2 1 bx 1 c

fsxd 5 ax 1 b

fsxd 5 a

a0anai

n

y 5 f sxd.

x

y

Up to

left

Up to

right

an > 0

Graphs of polynomial functions of even degree

The Leading Coefficient Test for polynomial functions

Figure P.29

x

y

Down

to left

Down

to right

an < 0

x

y

Down

to left

Up to

right

an > 0

Graphs of polynomial functions of odd degree

x

y

Up to

left

Down

to right

an < 0

an Þ 0fsxd 5 anxn 1 an21xn21 1 . . . 1 a2x2 1 a1x 1 a0,

FOR FURTHER INFORMATION For

more on the history of the concept of a

function, see the article “Evolution of

the Function Concept: A Brief Survey”

by Israel Kleiner in The College Mathe-

matics Journal. To view this article, go

to the website www.matharticles.com.

LEONHARD EULER (1707–1783)

In addition to making major contributions to

almost every branch of mathematics, Euler

was one of the first to apply calculus to

real-life problems in physics. His extensive

published writings include such topics as

shipbuilding, acoustics, optics, astronomy,

mechanics, and magnetism.

Bet

tman

n/C

orb

is


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Just as a rational number can be written as the quotient of two integers, a rational

function can be written as the quotient of two polynomials. Specifically, a function

is rational if it has the form

where and are polynomials.

Polynomial functions and rational functions are examples of algebraic

functions. An algebraic function of is one that can be expressed as a finite number

of sums, differences, multiples, quotients, and radicals involving For example,

is algebraic. Functions that are not algebraic are transcendental. For

instance, the trigonometric functions are transcendental.

Two functions can be combined in various ways to create new functions. For

example, given and you can form the functions shown.

Sum

Difference

Product

Quotient

You can combine two functions in yet another way, called composition. The

resulting function is called a composite function.

The composite of with may not be equal to the composite of with

EXAMPLE 4 Finding Composite Functions

Given and find each composite function.

a. b.

Solution

a. Definition of

Substitute cos

Definition of

Simplify.

b. Definition of

Substitute

Definition of

Note that s f 8 gdsxd Þ sg 8 f dsxd.

gsxd 5 coss2x 2 3d2x 2 3 for f sxd. 5 gs2x 2 3d

g 8 f sg 8 f dsxd 5 gs f sxdd 5 2 cos x 2 3

f sxd 5 2scos xd 2 3

x for gsxd. 5 fscos xdf 8 gs f 8 gdsxd 5 fsgsxdd

g 8 ff 8 g

gsxd 5 cos x,f sxd 5 2x 2 3

f.ggf

s fygdsxd 5fsxdgsxd

52x 2 3

x2 1 1

s fgdsxd 5 fsxdgsxd 5 s2x 2 3dsx2 1 1d s f 2 gdsxd 5 fsxd 2 gsxd 5 s2x 2 3d 2 sx2 1 1d s f 1 gdsxd 5 fsxd 1 gsxd 5 s2x 2 3d 1 sx2 1 1d

gsxd 5 x2 1 1,fsxd 5 2x 2 3

!x 1 1fsxd 5

xn.

x

qsxdpsxd

f

Domain of g

Domain of f

f

g

x

f (g(x))

g(x)

f g

The domain of the composite function

Figure P.30

f 8 g

qsxd Þ 0f sxd 5psxdqsxd

,

Definition of Composite Function

Let and be functions. The function given by is called

the composite of with The domain of is the set of all in the domain

of such that is in the domain of (see Figure P.30).fgsxdg

xf 8 gg.f

s f 8 gdsxd 5 fsgsxddgf


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In Section P.1, an -intercept of a graph was defined to be a point at which

the graph crosses the -axis. If the graph represents a function the number is a zero

of In other words, the zeros of a function are the solutions of the equation

For example, the function has a zero at because

In Section P.1 you also studied different types of symmetry. In the terminology of

functions, a function is even if its graph is symmetric with respect to the -axis, and

is odd if its graph is symmetric with respect to the origin. The symmetry tests in

Section P.1 yield the following test for even and odd functions.

NOTE Except for the constant function the graph of a function of cannot have

symmetry with respect to the -axis because it then would fail the Vertical Line Test for the

graph of the function.

EXAMPLE 5 Even and Odd Functions and Zeros of Functions

Determine whether each function is even, odd, or neither. Then find the zeros of the

function.

a. b.

Solution

a. This function is odd because

The zeros of are found as shown.

Let

Factor.

Zeros of

See Figure P.31(a).

b. This function is even because

The zeros of are found as shown.

Let

Subtract 1 from each side.

is an integer. Zeros of

See Figure P.31(b).

NOTE Each of the functions in Example 5 is either even or odd. However, some functions,

such as are neither even nor odd.f sxd 5 x2 1 x 1 1,

g x 5 s2n 1 1dp, n

cos x 5 21

gsxd 5 0. 1 1 cos x 5 0

g

coss2xd 5 cossxdgs2xd 5 1 1 coss2xd 5 1 1 cos x 5 gsxd.

f x 5 0, 1, 21

xsx2 2 1d 5 xsx 2 1dsx 1 1d 5 0

f sxd 5 0. x3 2 x 5 0

f

fs2xd 5 s2xd3 2 s2xd 5 2x3 1 x 5 2sx3 2 xd 5 2f sxd.

gsxd 5 1 1 cos xfsxd 5 x3 2 x

x

xf sxd 5 0,

y

fs4d 5 0.x 5 4f sxd 5 x 2 4

f sxd 5 0.ff.

af,x

sa, 0dx

x

−2 1 2

−2

−1

1

2

(1, 0)

(0, 0)

(−1, 0)f (x) = x3 − x

y

(a) Odd function

x

2 3 4π π ππ

2

3

1

−1

g(x) = 1 + cos x

y

(b) Even function

Figure P.31

Test for Even and Odd Functions

The function is even if

The function is odd if fs2xd 5 2fsxd.y 5 fsxdfs2xd 5 fsxd.y 5 fsxd

E X P L O R A T I O N

Use a graphing utility to graph each

function. Determine whether the

function is even, odd, or neither.

Describe a way to identify a function as

odd or even by inspecting the equation.

psxd 5 x9 1 3x 5 2 x 3 1 x

ksxd 5 x5 2 2x 4 1 x 2 2

jsxd 5 2 2 x6 2 x 8

hsxd 5 x5 2 2x3 1 x

gsxd 5 2x3 1 1

f sxd 5 x2 2 x4


! " # $ % # & ' ( ) ! * + , - ' # . % / % # .


In Exercises 1 and 2, use the graphs of and to answer the

following.

(a) Identify the domains and ranges of and

(b) Identify and

(c) For what value(s) of is

(d) Estimate the solution(s) of

(e) Estimate the solutions of

1. 2.

In Exercises 3–12, evaluate (if possible) the function at the given

value(s) of the independent variable. Simplify the results.

3. 4.

(a) (a)

(b) (b)

(c) (c)

(d) (d)

5. 6.

(a) (a)

(b) (b)

(c) (c)

(d) (d)

7. 8.

(a) (a)

(b) (b)

(c) (c)

9. 10.

11. 12.

In Exercises 13–18, find the domain and range of the function.

13. 14.

15. 16.

17. 18.

In Exercises 19-24, find the domain of the function.

19. 20.

21. 22.

23. 24.

In Exercises 25–28, evaluate the function as indicated.

Determine its domain and range.

25.

(a) (b) (c) (d)

26.

(a) (b) (c) (d)

27.

(a) (b) (c) (d)

28.

(a) (b) (c) (d)

In Exercises 29–36, sketch a graph of the function and find its

domain and range. Use a graphing utility to verify your graph.

29. 30.

31. 32.

33. 34.

35. 36. hsud 5 25 cos u

2gstd 5 2 sin pt

f sxd 5 x 1 !4 2 x2f sxd 5 !9 2 x2

f sxd 512x3 1 2hsxd 5 !x 2 1

gsxd 54

xf sxd 5 4 2 x

f s10df s5df s0df s23d

f sxd 5 5!x 1 4,

sx 2 5d2,

x ≤ 5

x > 5

f sb2 1 1df s3df s1df s23d

f sxd 5 5 |x| 1 1,

2x 1 1, x < 1

x ≥ 1

f ss2 1 2df s1df s0df s22d

f sxd 5 5x2 1 2,

2x2 1 2,

x ≤ 1

x > 1

f st2 1 1df s2df s0df s21d

f sxd 5 52x 1 1,

2x 1 2,

x < 0

x ≥ 0

gsxd 51

|x2 2 4|f sxd 51

|x 1 3|

hsxd 51

sin x 21

2

gsxd 52

1 2 cos x

f sxd 5 !x2 2 3x 1 2f sxd 5 !x 1 !1 2 x

gsxd 52

x 2 1f sxd 5

1

x

hstd 5 cot tf std 5 sec p t

4

gsxd 5 x2 2 5hsxd 5 2!x 1 3

f sxd 2 f s1dx 2 1

f sxd 2 f s2dx 2 2

f sxd 5 x3 2 xf sxd 51

!x 2 1

f sxd 2 f s1dx 2 1

f sx 1 Dxd 2 f sxdDx

f sxd 5 3x 2 1f sxd 5 x3

f s2py3df spy3df s5py4df s2py4df spdf s0d

f sxd 5 sin xf sxd 5 cos 2x

gst 1 4dgst 2 1dgscdgs22dgs3

2dgs!3 dgs4dgs0d

gsxd 5 x2sx 2 4dgsxd 5 3 2 x2

f sx 1 Dxdf sx 2 1df s25df sbdf s6df s23df s22df s0d

f sxd 5 !x 1 3f sxd 5 2x 2 3

y

x

2−2

2

4

4−4

f

g

y

x

−2

2

4

−4

4−4

fg

g xxc 5 0.

f xxc 5 2.

f xxc 5 g xxc?x

gx3c.f x22cg.f

gf

E x e r c i s e s f o r S e c t i o n P. 3 See www.CalcChat.com for worked-out solutions to odd-numbered exercises.

Writing About Concepts

37. The graph of the distance that a student drives in a

10-minute trip to school is shown in the figure. Give a verbal

description of characteristics of the student’s drive to school.

Time (in minutes)

Dis

tan

ce (

in m

iles

)

t

2 4 6 8 10

10

8

6

4

2

(0, 0)

(4, 2)

(6, 2)

(10, 6)

s


! " # $ % # & ' ( ) ! * + , - ' # . % / % # .


In Exercises 39–42, use the Vertical Line Test to determine

whether is a function of To print an enlarged copy of the

graph, go to the website www.mathgraphs.com.

39. 40.

41. 42.

In Exercises 43–46, determine whether is a function of

43. 44.

45. 46.

In Exercises 47–52, use the graph of to match the

function with its graph.

47. 48.

49. 50.

51. 52.

53. Use the graph of shown in the figure to sketch the graph of

each function. To print an enlarged copy of the graph, go to the

website www.mathgraphs.com.

(a) (b)

(c) (d)

(e) (f)

54. Use the graph of shown in the figure to sketch the graph of

each function. To print an enlarged copy of the graph, go to the

website www.mathgraphs.com.

(a) (b)

(c) (d)

(e) (f)

55. Use the graph of to sketch the graph of each

function. In each case, describe the transformation.

(a) (b) (c)

56. Specify a sequence of transformations that will yield each

graph of from the graph of the function

(a) (b)

57. Given and evaluate each expression.

(a) (b) (c)

(d) (e) (f)

58. Given and evaluate each expression.

(a) (b) (c)

(d) (e) (f)

In Exercises 59–62, find the composite functions and

What is the domain of each composite function? Are the

two composite functions equal?

59. 60.

61. 62.

63. Use the graphs of and to

evaluate each expression. If the

result is undefined, explain why.

(a) (b)

(c) (d)

(e) (f) f sgs21ddsg 8 f ds21ds f 8 gds23dgs f s5ddgs f s2dds f 8 gds3d

y

x

2−2

2

4−2

gf

gf

gsxd 5 !x 1 2gsxd 5 x2 2 1

f sxd 51

xf sxd 5

3

x

gsxd 5 cos xgsxd 5 !x

f sxd 5 x2 2 1f sxd 5 x2

xg 8 f c.x f 8 gc

gs f sxddf sgsxddg1 f 1p

422

gs f s0ddf 1g11

222f sgs2dd

gsxd 5 px,f sxd 5 sin x

gs f sxddf sgsxddf sgs24ddgs f s0ddgs f s1ddf sgs1dd

gsxd 5 x2 2 1,f sxd 5 !x

hsxd 5 2sinsx 2 1dhsxd 5 sin1x 1p

22 1 1

f sxd 5 sin x.h

y 5 !x 2 2y 5 2!xy 5 !x 1 2

f sxd 5 !x

12 f sxd2f sxdf sxd 2 1f sxd 1 4

(−4, −3)

−6

−5

4

(2, 1)

f

3f sx 1 2df sx 2 4d

f

14 f sxd3f sxdf sxd 2 4f sxd 1 2

−6

−7

9

3

f

f sx 2 1df sx 1 3d

f

y 5 f sx 2 1d 1 3y 5 f sx 1 6d 1 2

y 5 2f sx 2 4dy 5 2f s2xd 2 2

y 5 f sxd 2 5y 5 f sx 1 5d

y

x

1 2−1

−2

2

3

5

6

−3

−5

3−3 −2 4 5 7 9 10−4−6 −5

f (x)

ge

d

c

b a

y 5 f xxc

x2y 2 x2 1 4y 5 0y2 5 x2 2 1

x 2 1 y 5 4x2 1 y2 5 4

x.y

1

1

−1−1

x

y

2

1 2

1

−1

−2

−2

x

y

x2 1 y2 5 4y 5 5 x 1 1,

2x 1 2,

x ≤ 0

x > 0

x

−1

−2

−2−3

1

1

2

2

3

3

4

y

3 4

2

1 2

1

−1

−2

x

y

!x2 2 4 2 y 5 0x 2 y 2 5 0

x.y

Writing About Concepts (continued)

38. A student who commutes 27 miles to attend college

remembers, after driving a few minutes, that a term paper

that is due has been forgotten. Driving faster than usual, the

student returns home, picks up the paper, and once again

starts toward school. Sketch a possible graph of the

student’s distance from home as a function of time.


! " # $ % # & ' ( ) ! * + , - ' # . % / % # .


64. Ripples A pebble is dropped into a calm pond, causing

ripples in the form of concentric circles. The radius (in feet) of

the outer ripple is given by where is the time in

seconds after the pebble strikes the water. The area of the circle

is given by the function Find and interpret

Think About It In Exercises 65 and 66,

Identify functions for and (There are many correct

answers.)

65. 66.

In Exercises 67–70, determine whether the function is even,

odd, or neither. Use a graphing utility to verify your result.

67. 68.

69. 70.

Think About It In Exercises 71 and 72, find the coordinates of

a second point on the graph of a function f if the given point is

on the graph and the function is (a) even and (b) odd.

71. 72.

73. The graphs of and are shown in the figure. Decide

whether each function is even, odd, or neither.

Figure for 73 Figure for 74

74. The domain of the function shown in the figure is

(a) Complete the graph of given that is even.

(b) Complete the graph of given that is odd.

Writing Functions In Exercises 75–78, write an equation for a

function that has the given graph.

75. Line segment connecting and

76. Line segment connecting and

77. The bottom half of the parabola

78. The bottom half of the circle

Modeling Data In Exercises 79–82, match the data with a

function from the following list.

(i) (ii)

(iii) (iv)

Determine the value of the constant for each function such

that the function fits the data shown in the table.

79.

80.

81.

82.

83. Graphical Reasoning An electronically controlled thermo-

stat is programmed to lower the temperature during the night

automatically (see figure). The temperature in degrees

Celsius is given in terms of the time in hours on a 24-hour

clock.

(a) Approximate and

(b) The thermostat is reprogrammed to produce a temperature

How does this change the temperature?

Explain.

(c) The thermostat is reprogrammed to produce a temperature

How does this change the temperature?

Explain.

84. Water runs into a vase of height 30 centimeters at a constant

rate. The vase is full after 5 seconds. Use this information and

the shape of the vase shown to answer the questions if is the

depth of the water in centimeters and is the time in seconds

(see figure).

(a) Explain why is a function of

(b) Determine the domain and range of the function.

(c) Sketch a possible graph of the function.

30 cm

d

t.d

t

d

t

3 6 9 12 15 18 21 24

12

16

20

24

T

Hstd 5 Tstd 2 1.

Hstd 5 Tst 2 1d.

Ts15d.Ts4d

t,

T

c

rxxc 5 c/xhxxc 5 c!|x|gxxc 5 cx2f xxc 5 cx

x2 1 y2 5 4

x 1 y2 5 0

s5, 5ds1, 2ds0, 25ds24, 3d

ff

ff

26 ≤ x ≤ 6.

f

f

y

x

2

−4

−6

2

4

6

4 6−2−4−6

gh

f

y

x

2

4

4−4

hg,f,

s4, 9ds232, 4d

f sxd 5 sin2 xf sxd 5 x cos x

f sxd 5 3!xf sxd 5 x2s4 2 x2d

F sxd 5 24 sins1 2 xdF sxd 5 !2x 2 2

h.g,f,

Fxxc 5 f 8 g 8 h.

sA 8 rdstd.Asrd 5 pr 2.

trstd 5 0.6t,

x 0 1 4

y 0 2322222232

2124

x 0 1 4

y 0 1142

1421

2124

x 0 1 4

y Undef. 32 823228

2124

x 0 1 4

y 6 3 0 3 6

2124


! " # $ % # & ' ( ) ! * + , - ' # . % / % # .


85. Modeling Data The table shows the average numbers of acres

per farm in the United States for selected years. (Source:

U.S. Department of Agriculture)

(a) Plot the data where is the acreage and is the time in

years, with corresponding to 1950. Sketch a freehand

curve that approximates the data.

(b) Use the curve in part (a) to approximate

86. Automobile Aerodynamics The horsepower required to

overcome wind drag on a certain automobile is approximated by

where is the speed of the car in miles per hour.

(a) Use a graphing utility to graph

(b) Rewrite the power function so that represents the speed in

kilometers per hour. Find

87. Think About It Write the function

without using absolute value signs. (For a review of absolute

value, see Appendix D.)

88. Writing Use a graphing utility to graph the polynomial

functions and How many

zeros does each function have? Is there a cubic polynomial that

has no zeros? Explain.

89. Prove that the function is odd.

90. Prove that the function is even.

91. Prove that the product of two even (or two odd) functions is

even.

92. Prove that the product of an odd function and an even function

is odd.

93. Volume An open box of maximum volume is to be made

from a square piece of material 24 centimeters on a side by

cutting equal squares from the corners and turning up the sides

(see figure).

(a) Write the volume as a function of the length of the

corner squares. What is the domain of the function?

(b) Use a graphing utility to graph the volume function and

approximate the dimensions of the box that yield a maxi-

mum volume.

(c) Use the table feature of a graphing utility to verify your

answer in part (b). (The first two rows of the table are

shown.)

94. Length A right triangle is formed in the first quadrant by the

- and -axes and a line through the point (see figure).

Write the length of the hypotenuse as a function of

True or False? In Exercises 95–98, determine whether the

statement is true or false. If it is false, explain why or give an

example that shows it is false.

95. If then

96. A vertical line can intersect the graph of a function at most

once.

97. If for all in the domain of then the graph of

is symmetric with respect to the -axis.

98. If is a function, then f saxd 5 af sxd.f

yf

f,xf sxd 5 f s2xd

a 5 b.f sad 5 f sbd,

1 2 3 5 6 74

1

2

3

4

(3, 2)

x(x, 0)

(0, y)

y

x.L

s3, 2dyx

x,V

24 − 2x xx

x

24 − 2x

f sxd 5 a2n x2n 1 a2n22 x2n22 1 . . . 1 a2 x2 1 a0

f sxd 5 a2n11x2n11 1 . . . 1 a3 x3 1 a1x

p2sxd 5 x3 2 x.p1sxd 5 x3 2 x 1 1

f sxd 5 |x| 1 |x 2 2|

Hsxy1.6d.gfx

H.

x

10 ≤ x ≤ 100Hsxd 5 0.002x2 1 0.005x 2 0.029,

H

As15d.

t 5 0

tA

Year 1950 1960 1970 1980 1990 2000

Acreage 213 297 374 426 460 434

Length

Height, x and Width Volume, V

1

2 2f24 2 2s2dg 2 5 80024 2 2s2d

1f24 2 2s1dg 2 5 48424 2 2s1d

Putnam Exam Challenge

99. Let be the region consisting of the points of the

Cartesian plane satisfying both and

Sketch the region and find its area.

100. Consider a polynomial with real coefficients having the

property for every polynomial with real

coefficients. Determine and prove the nature of

These problems were composed by the Committee on the Putnam Prize Competition.

© The Mathematical Association of America. All rights reserved.

f sxd.gsxdf sgsxdd 5 gs f sxdd

f sxdR

|y| ≤ 1.|x| 2 |y| ≤ 1

sx, ydR


0 1 2 3 4 5 3 6 7 8 9 1 : ; < = 7 3 > 5 ? 5 3 >

SECTION P.4 Fitting Models to Data 31

Section P.4 Fitting Models to Data

• Fit a linear model to a real-life data set.

• Fit a quadratic model to a real-life data set.

• Fit a trigonometric model to a real-life data set.

Fitting a Linear Model to Data

A basic premise of science is that much of the physical world can be described

mathematically and that many physical phenomena are predictable. This scientific

outlook was part of the scientific revolution that took place in Europe during the late

1500s. Two early publications connected with this revolution were On the Revolutions

of the Heavenly Spheres by the Polish astronomer Nicolaus Copernicus and On the

Structure of the Human Body by the Belgian anatomist Andreas Vesalius. Each of

these books was published in 1543 and each broke with prior tradition by suggesting

the use of a scientific method rather than unquestioned reliance on authority.

One basic technique of modern science is gathering data and then describing the

data with a mathematical model. For instance, the data given in Example 1 are

inspired by Leonardo da Vinci’s famous drawing that indicates that a person’s height

and arm span are equal.

EXAMPLE 1 Fitting a Linear Model to Data

A class of 28 people collected the following data, which represent their heights and

arm spans (rounded to the nearest inch).

Find a linear model to represent these data.

Solution There are different ways to model these data with an equation. The

simplest would be to observe that and are about the same and list the model as

simply A more careful analysis would be to use a procedure from statistics

called linear regression. (You will study this procedure in Section 13.9.) The least

squares regression line for these data is

Least squares regression line

The graph of the model and the data are shown in Figure P.32. From this model, you

can see that a person’s arm span tends to be about the same as his or her height.

y 5 1.006x 2 0.23.

y 5 x.

yx

s67, 67ds71, 70d,s64, 63d,s65, 65d,s70, 72d,s69, 70d,s68, 67d,s71, 71d,s64, 64d,s63, 63d,s60, 61d,s69, 70d,s69, 68d,s70, 70d,s72, 73d,s62, 62d,s66, 68d,s65, 65d,s62, 60d,s71, 72d,s75, 74d,s70, 71d,s63, 63d,s61, 62d,s72, 73d,s68, 67d,s65, 65d,s60, 61d,

y

x

Height (in inches)

Arm

sp

an (

in i

nch

es)

60 62 64 66 68 70 72 74 76

60

76

x

74

72

70

68

66

64

62

y

Linear model and data

Figure P.32

A computer graphics drawing based on the

pen and ink drawing of Leonardo da Vinci’s

famous study of human proportions, called

Vitruvian Man

TECHNOLOGY Many scientific and graphing calculators have built-in least

squares regression programs. Typically, you enter the data into the calculator and

then run the linear regression program. The program usually displays the slope and

-intercept of the best-fitting line and the correlation coefficient The correlation

coefficient gives a measure of how well the model fits the data. The closer

is to 1, the better the model fits the data. For instance, the correlation coefficient for

the model in Example 1 is which indicates that the model is a good fit for

the data. If the -value is positive, the variables have a positive correlation, as in

Example 1. If the -value is negative, the variables have a negative correlation.r

r

r < 0.97,

|r|r.y


0 1 2 3 4 5 3 6 7 8 9 1 : ; < = 7 3 > 5 ? 5 3 >


Fitting a Quadratic Model to Data

A function that gives the height of a falling object in terms of the time is called a

position function. If air resistance is not considered, the position of a falling object can

be modeled by

where is the acceleration due to gravity, is the initial velocity, and is the initial

height. The value of depends on where the object is dropped. On earth, is approx-

imately feet per second per second, or meters per second per second.

To discover the value of experimentally, you could record the heights of a

falling object at several increments, as shown in Example 2.

EXAMPLE 2 Fitting a Quadratic Model to Data

A basketball is dropped from a height of about feet. The height of the basketball is

recorded 23 times at intervals of about 0.02 second.* The results are shown in the table.

Find a model to fit these data. Then use the model to predict the time when the

basketball will hit the ground.

Solution Begin by drawing a scatter plot of the data, as shown in Figure P.33. From

the scatter plot, you can see that the data do not appear to be linear. It does appear,

however, that they might be quadratic. To check this, enter the data into a calculator

or computer that has a quadratic regression program. You should obtain the model

Least squares regression quadratic

Using this model, you can predict the time when the basketball hits the ground by

substituting 0 for and solving the resulting equation for

Let

Quadratic Formula

Choose positive solution.

The solution is about 0.54 second. In other words, the basketball will continue to fall

for about 0.1 second more before hitting the ground.

t < 0.54

t 51.30 ± !s21.30d2 2 4s215.45ds5.234d

2s215.45d

s 5 0.0 5 215.45t2 2 1.30t 1 5.234

t.s

s 5 215.45t2 2 1.30t 1 5.234.

514

g

29.8232

gg

s0v0g

sstd 512gt2 1 v0t 1 s0

ts

Time (in seconds)

Hei

gh

t (i

n f

eet)

0.1 0.2 0.3 0.4 0.5

1

t

6

5

4

3

2

s

Scatter plot of data

Figure P.33

* Data were collected with a Texas Instruments CBL (Calculator-Based Laboratory) System.

Time 0.119996 0.139992 0.159988 0.179988 0.199984 0.219984

Height 4.85062 4.74979 4.63096 4.50132 4.35728 4.19523

Time 0.0 0.02 0.04 0.06 0.08 0.099996

Height 5.23594 5.20353 5.16031 5.0991 5.02707 4.95146

Time 0.23998 0.25993 0.27998 0.299976 0.319972 0.339961

Height 4.02958 3.84593 3.65507 3.44981 3.23375 3.01048

Time 0.359961 0.379951 0.399941 0.419941 0.439941

Height 2.76921 2.52074 2.25786 1.98058 1.63488


0 1 2 3 4 5 3 6 7 8 9 1 : ; < = 7 3 > 5 ? 5 3 >


Fitting a Trigonometric Model to Data

What is mathematical modeling? This is one of the questions that is asked in the book

Guide to Mathematical Modelling. Here is part of the answer.*

1. Mathematical modeling consists of applying your mathematical skills to obtain

useful answers to real problems.

2. Learning to apply mathematical skills is very different from learning mathematics

itself.

3. Models are used in a very wide range of applications, some of which do not appear

initially to be mathematical in nature.

4. Models often allow quick and cheap evaluation of alternatives, leading to optimal

solutions that are not otherwise obvious.

5. There are no precise rules in mathematical modeling and no “correct” answers.

6. Modeling can be learned only by doing.

EXAMPLE 3 Fitting a Trigonometric Model to Data

The number of hours of daylight on Earth depends on the latitude and the time of year.

Here are the numbers of minutes of daylight at a location of latitude on the

longest and shortest days of the year: June 21, 801 minutes; December 22, 655

minutes. Use these data to write a model for the amount of daylight (in minutes) on

each day of the year at a location of latitude. How could you check the accuracy

of your model?

Solution Here is one way to create a model. You can hypothesize that the model is

a sine function whose period is 365 days. Using the given data, you can conclude that

the amplitude of the graph is or 73. So, one possible model is

In this model, represents the number of each day of the year, with December 22

represented by A graph of this model is shown in Figure P.34. To check the

accuracy of this model, we used a weather almanac to find the numbers of minutes of

daylight on different days of the year at the location of latitude.

Dec 22 0 655 min 655 min

Jan 1 10 657 min 656 min

Feb 1 41 676 min 672 min

Mar 1 69 705 min 701 min

Apr 1 100 740 min 739 min

May 1 130 772 min 773 min

Jun 1 161 796 min 796 min

Jun 21 181 801 min 801 min

Jul 1 191 799 min 800 min

Aug 1 222 782 min 785 min

Sep 1 253 752 min 754 min

Oct 1 283 718 min 716 min

Nov 1 314 685 min 681 min

Dec 1 344 661 min 660 min

You can see that the model is fairly accurate.

Daylight Given by ModelActual DaylightValue of tDate

208 N

t 5 0.

t

d 5 728 2 73 sin12p t

3651

p

22.

s801 2 655dy2,

208 N

d

208 N

* Text from Dilwyn Edwards and Mike Hamson, Guide to Mathematical Modelling (Boca Raton:

CRC Press, 1990). Used by permission of the authors.

Day (0 ↔ December 22)

Day

light

(in m

inute

s)

40 120 200 280 360 440

650

t

850

800

750

700

73

73

728

365

d

Graph of model

Figure P.34

The plane of Earth’s orbit about the sun and

its axis of rotation are not perpendicular.

Instead, Earth’s axis is tilted with respect

to its orbit. The result is that the amount of

daylight received by locations on Earth

varies with the time of year. That is, it varies

with the position of Earth in its orbit.

NOTE For more review of trigonometric

functions, see Appendix D.


0 1 2 3 4 5 3 6 7 8 9 1 : ; < = 7 3 > 5 ? 5 3 >


In Exercises 1–4, a scatter plot of data is given. Determine

whether the data can be modeled by a linear function, a quadratic

function, or a trigonometric function, or that there appears to be

no relationship between and To print an enlarged copy of the

graph, go to the website www.mathgraphs.com.

1. 2.

3. 4.

5. Carcinogens Each ordered pair gives the exposure index of

a carcinogenic substance and the cancer mortality per 100,000

people in the population.

(a) Plot the data. From the graph, do the data appear to be

approximately linear?

(b) Visually find a linear model for the data. Graph the model.

(c) Use the model to approximate if

6. Quiz Scores The ordered pairs represent the scores on two

consecutive 15-point quizzes for a class of 18 students.

(a) Plot the data. From the graph, does the relationship between

consecutive scores appear to be approximately linear?

(b) If the data appear to be approximately linear, find a linear

model for the data. If not, give some possible explanations.

7. Hooke’s Law Hooke’s Law states that the force required to

compress or stretch a spring (within its elastic limits) is propor-

tional to the distance that the spring is compressed or stretched

from its original length. That is, where is a measure of

the stiffness of the spring and is called the spring constant. The

table shows the elongation in centimeters of a spring when a

force of newtons is applied.

(a) Use the regression capabilities of a graphing utility to find a

linear model for the data.

(b) Use a graphing utility to plot the data and graph the model.

How well does the model fit the data? Explain your

reasoning.

(c) Use the model to estimate the elongation of the spring when

a force of 55 newtons is applied.

8. Falling Object In an experiment, students measured the speed

(in meters per second) of a falling object seconds after it was

released. The results are shown in the table.


linear model for the data.



reasoning.

(c) Use the model to estimate the speed of the object after

2.5 seconds.

9. Energy Consumption and Gross National Product The data

show the per capita electricity consumptions (in millions of Btu)

and the per capita gross national products (in thousands of U.S.

dollars) for several countries in 2000. (Source: U.S. Census

Bureau)


linear model for the data. What is the correlation coefficient?


(c) Interpret the graph in part (b). Use the graph to identify the

three countries that differ most from the linear model.

(d) Delete the data for the three countries identified in part

(c). Fit a linear model to the remaining data and give the

correlation coefficient.

ts

F

d

kF 5 kd,

d

F

s9, 6ds10, 15d,s14, 11d,s11, 10d,s7, 14d,s11, 14d,s9, 10d,s10, 11d,s15, 9d,s8, 10d,s14, 15d,s14, 11d,

s9, 7d,s10, 15d,s15, 15d,s14, 14d,s9, 7d,s7, 13d,

x 5 3.y

s9.35, 213.4ds7.42, 181.0d,s12.65, 210.7d,s4.85, 165.5d,s2.63, 140.7d,s2.26, 116.7d,s4.42, 132.9d,s3.58, 133.1d,s3.50, 150.1d,

y

x

x

y

x

y

x

y

x

y

y.x

E x e r c i s e s f o r S e c t i o n P. 4 See www.CalcChat.com for worked-out solutions to odd-numbered exercises.

20 40 60 80 100

1.4 2.5 4.0 5.3 6.6d

F

0 1 2 3 4

0 11.0 19.4 29.2 39.4s

t

Argentina (73, 12.05)

Chile (68, 9.1)

Greece (126, 16.86)

Hungary (105, 11.99)

Mexico (63, 8.79)

Portugal (108, 16.99)

Spain (137, 19.26)

United Kingdom (166, 23.55)

Bangladesh (4, 1.59)

Egypt (32, 3.67)

Hong Kong (118, 25.59)

India (13, 2.34)

Poland (95, 9)

South Korea (167, 17.3)

Turkey (47, 7.03)

Venezuela (113, 5.74)


0 1 2 3 4 5 3 6 7 8 9 1 : ; < = 7 3 > 5 ? 5 3 >


10. Brinell Hardness The data in the table show the Brinell

hardness of 0.35 carbon steel when hardened and tempered

at temperature (degrees Fahrenheit). (Source: Standard

Handbook for Mechanical Engineers)

(a) Use the regression capabilities of a graphing utility to find

a linear model for the data.



reasoning.

(c) Use the model to estimate the hardness when is 500 F.

11. Automobile Costs The data in the table show the variable

costs for operating an automobile in the United States for

several recent years. The functions and represent the

costs in cents per mile for gas and oil, maintenance,

and tires, respectively. (Source: American Automobile

Manufacturers Association)


a cubic model for and linear models for and

(b) Use a graphing utility to graph and

in the same viewing window. Use the model to estimate the

total variable cost per mile in year 12.

12. Beam Strength Students in a lab measured the breaking

strength (in pounds) of wood 2 inches thick, inches high,

and 12 inches long. The results are shown in the table.

(a) Use the regression capabilities of a graphing utility to fit a

quadratic model to the data.


(c) Use the model to approximate the breaking strength when

13. Health Maintenance Organizations The bar graph shows

the numbers of people (in millions) receiving care in HMOs

for the years 1990 through 2002. (Source: Centers for

Disease Control)

(a) Let be the time in years, with corresponding to

1990. Use the regression capabilities of a graphing utility to

find linear and cubic models for the data.

(b) Use a graphing utility to graph the data and the linear and

cubic models.

(c) Use the graphs in part (b) to determine which is the better

model.

(d) Use a graphing utility to find and graph a quadratic model

for the data.

(e) Use the linear and cubic models to estimate the number of

people receiving care in HMOs in the year 2004.

(f) Use a graphing utility to find other models for the data.

Which models do you think best represent the data?

Explain.

14. Car Performance The time (in seconds) required to attain a

speed of miles per hour from a standing start for a Dodge

Avenger is shown in the table. (Source: Road & Track)


a quadratic model for the data.


(c) Use the graph in part (b) to state why the model is not

appropriate for determining the times required to attain

speeds less than 20 miles per hour.

(d) Because the test began from a standing start, add the point

to the data. Fit a quadratic model to the revised data

and graph the new model.

(e) Does the quadratic model more accurately model the

behavior of the car for low speeds? Explain.

s0, 0d

s

t

t 5 0t

Year (0 ↔ 1990)

Enro

llm

ent

(in m

illi

ons)

0 1 2 3 4 5 6 7 8 9 10 11 12

t

90

80

70

60

50

40

30

20

10

N

HMO Enrollment

34.0 36.142.2

50.9 52.5

66.8

76.681.3 80.9 79.5

76.1

33.038.4

N

x 5 2.

xS

y1 1 y2 1 y3y3,y2,y1,

y3.y2y1

y3y2,y1,

8t

t

H

t 200 400 600 800 1000 1200

H 534 495 415 352 269 217

x 4 6 8 10 12

S 2370 5460 10,310 16,250 23,860

s 30 40 50 60 70 80 90

t 3.4 5.0 7.0 9.3 12.0 15.8 20.0

Year

0 5.40 2.10 0.90

1 6.70 2.20 0.90

2 6.00 2.20 0.90

3 6.00 2.40 0.90

4 5.60 2.50 1.10

5 6.00 2.60 1.40

6 5.90 2.80 1.40

7 6.60 2.80 1.40

y3y2y1


0 1 2 3 4 5 3 6 7 8 9 1 : ; < = 7 3 > 5 ? 5 3 >


15. Car Performance A V8 car engine is coupled to a dynamome-

ter and the horsepower is measured at different engine speeds

(in thousands of revolutions per minute). The results are

shown in the table.


a cubic model for the data.


(c) Use the model to approximate the horsepower when the

engine is running at 4500 revolutions per minute.

16. Boiling Temperature The table shows the temperatures

at which water boils at selected pressures (pounds per

square inch). (Source: Standard Handbook for Mechanical

Engineers)


a cubic model for the data.


(c) Use the graph to estimate the pressure required for the boil-

ing point of water to exceed 300 F.

(d) Explain why the model would not be correct for pressures

exceeding 100 pounds per square inch.

17. Harmonic Motion The motion of an oscillating weight

suspended by a spring was measured by a motion detector. The

data collected and the approximate maximum (positive and

negative) displacements from equilibrium are shown in the

figure. The displacement is measured in centimeters and the

time is measured in seconds.

(a) Is a function of Explain.

(b) Approximate the amplitude and period of the oscillations.

(c) Find a model for the data.

(d) Use a graphing utility to graph the model in part (c).

Compare the result with the data in the figure.

18. Temperature The table shows the normal daily high tempera-

tures for Honolulu and Chicago (in degrees Fahrenheit) for

month with corresponding to January. (Source: NOAA)

(a) A model for Honolulu is

Find a model for Chicago.

(b) Use a graphing utility to graph the data and the model for

the temperatures in Honolulu. How well does the model fit?

(c) Use a graphing utility to graph the data and the model for

the temperatures in Chicago. How well does the model fit?

(d) Use the models to estimate the average annual temperature

in each city. What term of the model did you use? Explain.

(e) What is the period of each model? Is it what you expected?

Explain.

(f) Which city has a greater variability of temperatures

throughout the year? Which factor of the models deter-

mines this variability? Explain.

Hstd 5 84.40 1 4.28 sin1pt

61 3.862.

t 5 1t,

CH

3

2

1

−1

t

0.2 0.4 0.6 0.8

(0.125, 2.35)

(0.375, 1.65)

y

t?y

t

y

8

p

s8FdT

x

y

t 1 2 3 4 5 6

H 80.1 80.5 81.6 82.8 84.7 86.5

C 29.0 33.5 45.8 58.6 70.1 79.6

t 7 8 9 10 11 12

H 87.5 88.7 88.5 86.9 84.1 81.2

C 83.7 81.8 74.8 63.3 48.4 34.0

p 30 40 60 80 100

T 327.818312.038292.718267.258250.338

p 5 10 (1 atmosphere) 20

T 227.968212.008193.218162.248

14.696

x 1 2 3 4 5 6

y 40 85 140 200 225 245

Writing About Concepts

19. Search for real-life data in a newspaper or magazine. Fit the

data to a model. What does your model imply about the data?

20. Describe a possible real-life situation for each data set.

Then describe how a model could be used in the real-life

setting.

(a) (b)

(c) (d)

x

y

x

y

x

y

x

y


0 1 2 3 4 5 3 6 7 8 9 1 : ; < = 7 3 > 5 ? 5 3 >

REVIEW EXERCISES 37

In Exercises 1–4, find the intercepts (if any).

1. 2.

3. 4.

In Exercises 5 and 6, check for symmetry with respect to both

axes and to the origin.

5. 6.

In Exercises 7–14, sketch the graph of the equation.

7. 8.

9. 10.

11. 12.

13. 14.

In Exercises 15 and 16, describe the viewing window of a graph-

ing utility that yields the figure.

15. 16.

In Exercises 17 and 18, use a graphing utility to find the point(s)

of intersection of the graphs of the equations.

17. 18.

19. Think About It Write an equation whose graph has

intercepts at and and is symmetric with respect

to the origin.

20. Think About It For what value of does the graph of

pass through the point?

(a) (b) (c) (d)

In Exercises 21 and 22, plot the points and find the slope of the

line passing through the points.

21. 22.

In Exercises 23 and 24, use the concept of slope to find t such

that the three points are collinear.

23. 24.

In Exercises 25–28, find an equation of the line that passes

through the point with the indicated slope. Sketch the line.

25. 26.

27. 28. is undefined.

29. Find equations of the lines passing through and having

the following characteristics.

(a) Slope of

(b) Parallel to the line

(c) Passing through the origin

(d) Parallel to the -axis

30. Find equations of the lines passing through and having

the following characteristics.

(a) Slope of

(b) Perpendicular to the line

(c) Passing through the point

(d) Parallel to the -axis

31. Rate of Change The purchase price of a new machine is

$12,500, and its value will decrease by $850 per year. Use this

information to write a linear equation that gives the value of

the machine years after it is purchased. Find its value at the

end of 3 years.

32. Break-Even Analysis A contractor purchases a piece of

equipment for $36,500 that costs an average of $9.25 per hour

for fuel and maintenance. The equipment operator is paid

$13.50 per hour, and customers are charged $30 per hour.

(a) Write an equation for the cost of operating this equip-

ment for hours.

(b) Write an equation for the revenue derived from hours

of use.

(c) Find the break-even point for this equipment by finding the

time at which

In Exercises 33–36, sketch the graph of the equation and use the

Vertical Line Test to determine whether the equation expresses

as a function of

33. 34.

35. 36.

37. Evaluate (if possible) the function at the specified

values of the independent variable, and simplify the results.

(a) (b)

38. Evaluate (if possible) the function at each value of the indepen-

dent variable.

(a) (b) (c)

39. Find the domain and range of each function.

(a) (b) (c) y 5 5x2,

2 2 x, x < 0

x ≥ 0y 5

7

2x 2 10y 5 !36 2 x2

f s1df s0df s24d

f sxd 5 5 x2 1 2,

|x 2 2|, x < 0

x ≥ 0

f s1 1 Dxd 2 f s1dDx

f s0d

f sxd 5 1yx

x 5 9 2 y 2y 5 x2 2 2x

x2 2 y 5 0x 2 y 2 5 0

x.y

R 5 C.

tR

t

C

t

V

x

s2, 4dx 1 y 5 0

223

s1, 3dy

5x 2 3y 5 3

716

s22, 4d

ms5, 4d,m 5 223s23, 0d,

m 5 0s22, 6d,m 532s0, 25d,

s23, 3d, st, 21d, s8, 6ds22, 5d, s0, td, s1, 1d

s7, 21d, s7, 12ds32, 1d, s5,

52d

s21, 21ds0, 0ds22, 1ds1, 4d

y 5 kx3k

x 5 2x 5 22

y 2 x2 5 7 x 1 y 5 5

x 2 y 1 1 5 03x 2 4y 5 8

y 5 8 3!x 2 6y 5 4x2 2 25

y 5 |x 2 4| 2 4y 5 !5 2 x

y 5 6x 2 x2y 5 7 2 6x 2 x2

0.02x 1 0.15y 5 0.25213x 1

56y 5 1

4x 2 2y 5 6y 512s2x 1 3d

y 5 x4 2 x2 1 3x2y 2 x2 1 4y 5 0

xy 5 4y 5x 2 1

x 2 2

y 5 sx 2 1dsx 2 3dy 5 2x 2 3

R e v i e w E x e r c i s e s f o r C h a p t e r P See www.CalcChat.com for worked-out solutions to odd-numbered exercises.


@ A B C D E C F G H I A J K L M G C N E O E C N


40. Given and evaluate each

expression.

(a) (b) (c)

41. Sketch (on the same set of coordinate axes) graphs of for

and 2.

(a) (b)

(c) (d)

42. Use a graphing utility to graph Use the graph

to write a formula for the function shown in the figure.

To print an enlarged copy of the graph, go to the website

www.mathgraphs.com.

(a) (b)

43. Conjecture

(a) Use a graphing utility to graph the functions and in

the same viewing window. Write a description of any

similarities and differences you observe among the graphs.

Odd powers:

Even powers:

(b) Use the result in part (a) to make a conjecture about the

graphs of the functions and Use a graphing

utility to verify your conjecture.

44. Think About It Use the result of Exercise 43 to guess the

shapes of the graphs of the functions and Then use a

graphing utility to graph each function and compare the result

with your guess.

(a) (b)

(c)

45. Area A wire 24 inches long is to be cut into four pieces to

form a rectangle whose shortest side has a length of

(a) Write the area of the rectangle as a function of

(b) Determine the domain of the function and use a graphing

utility to graph the function over that domain.

(c) Use the graph of the function to approximate the maximum

area of the rectangle. Make a conjecture about the dimen-

sions that yield a maximum area.

46. Writing The following graphs give the profits for two small

companies over a period of 2 years. Create a story to describe

the behavior of each profit function for some hypothetical

product the company produces.

(a) (b)

47. Think About It What is the minimum degree of the polyno-

mial function whose graph approximates the given graph?

What sign must the leading coefficient have?

(a) (b)

(c) (d)

48. Stress Test A machine part was tested by bending it

centimeters 10 times per minute until the time (in hours) of

failure. The results are recorded in the table.


a linear model for the data.


(c) Use the graph to determine whether there may have been an

error made in conducting one of the tests or in recording the

results. If so, eliminate the erroneous point and find the

model for the remaining data.

49. Harmonic Motion The motion of an oscillating weight

suspended by a spring was measured by a motion detector. The

data collected and the approximate maximum (positive and

negative) displacements from equilibrium are shown in the

figure. The displacement is measured in feet and the time is

measured in seconds.

(a) Is a function of Explain.

(b) Approximate the amplitude and period of the oscillations.

(c) Find a model for the data.

(d) Use a graphing utility to graph the model in part (c).

Compare the result with the data in the figure.

y

0.50

0.25

−0.25

−0.50

t

(1.1, 0.25)

(0.5, −0.25)

1.0 2.0

t?y

ty

y

x

y

x

−4 2 4

4

2

−4

y

x

−2 2

2

4

−4

−2

y

x

−4 2

2

4

−6

y

x

−4 −2 2 4

4

−2

−4

100,000

50,000

p

P

1 2

200,000

100,000

p

P

1 2

p

P

x.A

x.

hsxd 5 x3sx 2 6d3

gsxd 5 x3sx 2 6d2f sxd 5 x2sx 2 6d2

h.g,f,

y 5 x8.y 5 x7

hsxd 5 x6gsxd 5 x4,f sxd 5 x2,

hsxd 5 x5gsxd 5 x3,f sxd 5 x,

hg,f,

−1

−4

2

6

(2, 1)

(4, −3)

g

−2

−1

4

(2, 5)

(0, 1)

6

g

g

f sxd 5 x3 2 3x2.

f sxd 5 cx3f sxd 5 sx 2 2d3 1 c

f sxd 5 sx 2 cd3f sxd 5 x3 1 c

c 5 22, 0,

f

gs f sxddf sxdgsxdf sxd 2 gsxd

gsxd 5 2x 1 1,f sxd 5 1 2 x2

x 3 6 9 12 15 18 21 24 27 30

y 61 56 53 55 48 35 36 33 44 23


@ A B C D E C F G H I A J K L M G C N E O E C N

P.S. Problem Solving 39

1. Consider the circle as shown in the

figure.

(a) Find the center and radius of the circle.

(b) Find an equation of the tangent line to the circle at the point

(c) Find an equation of the tangent line to the circle at the point

(d) Where do the two tangent lines intersect?


2. There are two tangent lines from the point to the circle

(see figure). Find equations of these two lines

by using the fact that each tangent line intersects the circle in

exactly one point.

3. The Heaviside function is widely used in engineering

applications.

Sketch the graph of the Heaviside function and the graphs of the

following functions by hand.

(a) (b) (c)

(d) (e) (f)

4. Consider the graph of the function shown below. Use

this graph to sketch the graphs of the following functions.

To print an enlarged copy of the graph, go to the website

www.mathgraphs.com.

(a) (b) (c) (d)

(e) (f) (g)

5. A rancher plans to fence a rectangular pasture adjacent to a river.

The rancher has 100 meters of fence, and no fencing is needed

along the river (see figure).

(a) Write the area of the pasture as a function of the length

of the side parallel to the river. What is the domain of

(b) Graph the area function and estimate the dimensions

that yield the maximum amount of area for the pasture.

(c) Find the dimensions that yield the maximum amount of area

for the pasture by completing the square.


6. A rancher has 300 feet of fence to enclose two adjacent pastures.

(a) Write the total area of the two pastures as a function of

(see figure). What is the domain of

(b) Graph the area function and estimate the dimensions that

yield the maximum amount of area for the pastures.

(c) Find the dimensions that yield the maximum amount of area

for the pastures by completing the square.

7. You are in a boat 2 miles from the nearest point on the coast. You

are to go to a point located 3 miles down the coast and

1 mile inland (see figure). You can row at 2 miles per hour and

walk at 4 miles per hour. Write the total time of the trip as a

function of

Q

2 mi

x 3 − x

3 mi

1 mi

x.

T

Q

A?

xA

xxx

yy

y

x

y

AsxdA?

x,A

x

2

−1

−2

−2

1

y

f

f s|x|d| f sxd|2f sxdf s2xd2 f sxdf sxd 1 1f sx 1 1d

f

2Hsx 2 2d 1 212 HsxdHs2xd

2HsxdHsx 2 2dHsxd 2 2

Hsxd 5 51,

0,

x ≥ 0

x < 0

Hsxd

x2 1 sy 1 1d2 5 1

s0, 1d

x

32−3

−3

−2

−4

−1

1

2

y

x

86−2

−2

2

4

6

8

y

s6, 0d.

s0, 0d.

x2 1 y2 2 6x 2 8y 5 0,

P.S. Problem Solving See www.CalcChat.com for worked-out solutions to odd-numbered exercises.

OLIVER HEAVISIDE (1850–1925)

Heaviside was a British mathematician and physicist who contributed to the

field of applied mathematics, especially applications of mathematics to

electrical engineering. The Heaviside function is a classic type of “on-off ”

function that has applications to electricity and computer science.

Inst

itute

of

Ele

ctri

cal

Engin

eers

,L

ondon


P Q R S T U S V W X Y Q Z [ \ ] W S ^ U _ U S ^


8. You drive to the beach at a rate of 120 kilometers per hour. On

the return trip, you drive at a rate of 60 kilometers per hour. What

is your average speed for the entire trip? Explain your reasoning.

9. One of the fundamental themes of calculus is to find the slope

of the tangent line to a curve at a point. To see how this can be

done, consider the point on the graph of

(a) Find the slope of the line joining and Is the

slope of the tangent line at greater than or less than

this number?

(b) Find the slope of the line joining and Is the


this number?

(c) Find the slope of the line joining and Is

the slope of the tangent line at greater than or less

than this number?

(d) Find the slope of the line joining and

in terms of the nonzero number Verify that

and 0.1 yield the solutions to parts (a)–(c) above.

(e) What is the slope of the tangent line at Explain how

you arrived at your answer.

10. Sketch the graph of the function and label the point

on the graph.

(a) Find the slope of the line joining and Is the


this number?

(b) Find the slope of the line joining and . Is the


this number?

(c) Find the slope of the line joining and Is

the slope of the tangent line at greater than or less

than this number?

(d) Find the slope of the line joining and

in terms of the nonzero number

(e) What is the slope of the tangent line at the point

Explain how you arrived at your answer.

11. A large room contains two speakers that are 3 meters apart. The

sound intensity of one speaker is twice that of the other, as

shown in the figure. (To print an enlarged copy of the graph, go

to the website www.mathgraphs.com.) Suppose the listener is

free to move about the room to find those positions that receive

equal amounts of sound from both speakers. Such a

location satisfies two conditions: (1) the sound intensity at the

listener’s position is directly proportional to the sound level of

a source, and (2) the sound intensity is inversely proportional to

the square of the distance from the source.

(a) Find the points on the -axis that receive equal amounts of

sound from both speakers.

(b) Find and graph the equation of all locations where

one could stand and receive equal amounts of sound from

both speakers.


12. Suppose the speakers in Exercise 11 are 4 meters apart and the

sound intensity of one speaker is times that of the other, as

shown in the figure. To print an enlarged copy of the graph, go

to the website www.mathgraphs.com.

(a) Find the equation of all locations where one could stand

and receive equal amounts of sound from both speakers.

(b) Graph the equation for the case

(c) Describe the set of locations of equal sound as becomes

very large.

13. Let and be the distances from the point to the points

and respectively, as shown in the figure. Show

that the equation of the graph of all points satisfying

is This curve is called a

lemniscate. Graph the lemniscate and identify three points on

the graph.

14. Let

(a) What are the domain and range of

(b) Find the composition What is the domain of this

function?

(c) Find What is the domain of this function?

(d) Graph Is the graph a line? Why or why not?f s f s f sxddd.f s f s f sxddd.

f s f sxdd.f?

f sxd 51

1 2 x.

1

1

−1

−1x

d2

d1

(x, y)

y

sx2 1 y2d2 5 2sx2 2 y2d.d1d2 5 1

sx, yds1, 0d,s21, 0d

sx, ydd2d1

k

k 5 3.

sx, yd

k

x

431 2

I kI

1

2

3

4

y

x

31 2

I 2I

1

2

3

y

sx, yd

x

I

s4, 2d?h.f s4 1 hdd

s4 1 h,s4, 2d

s4, 2ds4.41, 2.1d.s4, 2d

s4, 2ds1, 1ds4, 2d

s4, 2ds9, 3d.s4, 2d

s4, 2df sxd 5 !x

s2, 4d?21,h 5 1,

h.f s2 1 hdds2 1 h,s2, 4d

s2, 4ds2.1, 4.41d.s2, 4d

s2, 4ds1, 1d.s2, 4d

s2, 4ds3, 9d.s2, 4d

x

62 4−6 −2−4

2

4

6

8

10

(2, 4)

y

f sxd 5 x2.s2, 4d



Chapter P

Section P.1 (page 8)

1. b 2. d 3. a 4. c

5. Answers will vary. 7. Answers will vary.

9. Answers will vary.



15. 17.

(a) (b)

19. 21.

23. 25. 27. Symmetric with respect to the axis

29. Symmetric with respect to the axis

31. Symmetric with respect to the origin 33. No symmetry

35. Symmetric with respect to the origin

37. Symmetric with respect to the axis

39. 41.

Symmetry: none Symmetry: none

43. 45.

Symmetry: axis Symmetry: none

47. 49.

Symmetry: none Symmetry: none

51. 53.

Symmetry: origin Symmetry: originy

3

1

2

x321

x1

−2

−3

−4

2

3

4

−2 −1−3−4 2 3 4

(0, 0)

y

y 5 1yxx 5 y3

x

1

2

3

4

5

−2

6

−4 −3 −1 41 2 3

(−2, 0) (0, 0)

yy

5

4

3

3

x32

,

1

1

1

,,

23

)02 )(0 2(

y 5 x!x 1 2y 5 x3 1 2

x

2

−2

8

10

12

−8 −6−10 2 4(−3, 0)

(0, 9)

y

x)0(1,

0, 1)

y

2

),( 1 0

(

1

2

2 2−

y-

y 5 sx 1 3d2y 5 1 2 x2

x108

)0,8(

y

42

)4,

2

(

2

0

2

8

6

10

0

y

)20( ,2

1

x321

1

,23

y 512 x 2 4y 5 23x 1 2

y-

x-

y-s0, 0ds4, 0ds0, 0d, s5, 0d, s25, 0ds0, 22d, s22, 0d, s1, 0d

x 5 24.00y < 1.73

−6 6

−3

5

( 4.00, 3)−

(2, 1.73)

y 5 !5 2 xXmin = -3

Xmax = 5

Xscl = 1

Ymin = -3

Ymax = 5

Yscl = 1

−1−2 1 2 3−1

1

2

3

x

(1, 2)

(2, 1)

(−2, −1)

(−1, −2)

−3, − ) 2

3) ) 2

33, )

y

x

4

6

8

2

−6

−8

−4

−10

10

−2 2 1614 1812

(0, −4)

(1, −3)

(4, −2) (16, 0)

y

(9, −1)

x

2

−2

4

6

−4−6 2

(−3, 1) (−1, 1)

(−4, 2)

(−2, 0)

(0, 2)

(1, 3)(−5, 3)

y

x

2

−4

−2

−6

6

−4−6 4 6

(−3, −5) (3, −5)

(−2, 0)

(0, 4)

(2, 0)

y

x2

2

−4

−6

−8

4

6

8

−4−6−8 4 6 8

(−4, −5)(−2, −2)

(0, 1)

(2, 4)

(4, 7)

y

A25

Answers to Odd-Numbered Exercises

0 2 4

1 4 72225y

2224x 0 2 3

0 4 0 2525y

2223x

0 1

3 2 1 0 1 2 3y

2122232425x

0 1 4 9 16

021222324y

x

0 1 2 3

Undef. 2 1 2322212

23y

212223x



55. 57.

Symmetry: axis

Symmetry: axis

59.

Symmetry: axis

61. 63. 65.

67.

69. 71.

73. (a)

(b) (c) 217

75. units 77.

79. (i) b; (ii) d; (iii) a; (iv) c;

81. False. is not a point on the graph of

83. True 85.


1. 3. 5.

7. 9.

11. is undefined. 13.

15. 17.

19. (a) (b) ft

21. (a) (b) Population increased least

rapidly: 2000 –2001

23. 25. is undefined, no intercept

27. 29.

31. 33.

35. 37.

39. 41.

43. 45. 47.

49. 51.

x

y

−2 −1 1 2

−1

3

x

y

−1−2−3 1 2 3 4 5

−2

−4

−5

−6

1

2

x 1 y 2 3 5 03x 1 2y 2 6 5 0x 2 3 5 0

x1

2

1

3

4

−2 −1−3−4 2 3 4

( )12

72

,

( )34

0,

y

1

2

3

4

5

−2

6

7

8

9

x−1 4 6 7 8 91 2 3

(5, 1)

(5, 8)

y

22x 2 4y 1 3 5 0x 2 5 5 0

1

2

3

4

5

−2

6

7

8

9

x−1 4 6 7 8 91 2 3

(5, 0)

(2, 8)

y

x

y

−2 −1 2 3 4 5−1

−2

−3

−5

1

2

(2, 1)

(0, 3)−

8x 1 3y 2 40 5 02x 2 y 2 3 5 0

x2

2

−8

4

6

8

−4 −2−6−8 4 6 8

(2, 6)

(0, 0)

y

x

1

2

−3

−1

−2

−4

−5

3

−1−2 1 2 3 4 5 6

(3, −2)

y

3x 2 y 5 03x 2 y 2 11 5 0

x

y

1 2 3 4

−1

2

3

4

(0, 0)

x

y

1−1−2−3−4

1

2

4

5

(0, 3)

2x 2 3y 5 03x 2 4y 1 12 5 0

y-mm 5 215, s0, 4d

tPopula

tion (

in m

illi

ons)

Year (6 ↔ 1996)

6 7 8 9 10 11

260

270

280

290

y

10!1013

s0, 10d, s2, 4d, s3, 1ds0, 1d, s1, 1d, s3, 1d

x

−1

−2

−3

2

3

−2−3 21 3

y

( )12

23

,− ( )34

16

,−

x

1

2

−1

−2

3

4

5

6

−1−2 1 3 4 5 6

(2, 1)

(2, 5)

y

m 5 2m

x

y

1 2 3 5 6 7−1

(3, 4)−

(5, 2)

−2

−3

−4

−5

1

2

3

m 5 3

2

3

4

5

−1

1

x4 51 3

(2, 3)

m = −2

m = 1

32

m = −m is

undefined.

y

m 5 212m 5 0m 5 1

x2 1 sy 2 4d2 5 4

x 514 y2.s21, 22d

k 5 36k 5 3k 5 210k 5 2

y 5 sx 1 2dsx 2 4dsx 2 6dx < 3133

35

−50

−5

250

y 5 20.007t2 1 4.82t 1 35.4

s23, !3ds22, 2d,s21, 25d, s0, 21d, s2, 1ds21, 21d, s0, 0d, s1, 1d

s21, 22d, s2, 1ds21, 5d, s2, 2ds1, 1dx-

y2 5 2!6 2 x

3

8

−3

−1

3

(6, 0)

( )0, 2

( )0, − 2

y1 5!6 2 x

3

1

−4

−11

4

(−9, 0)

(0, −3)

(0, 3)

x-

x2

2

−4

−2

−6

−8

4

6

8

−4 −2−8 4 6 8

(−6, 0)

(0, 6)

(6, 0)

y

y2 5 2!x 1 9y-

y1 5 !x 1 9y 5 6 2 |x|

A26 Answers to Odd-Numbered Exercises


` a b c d e c f g h i a j k l m g c n e o e c n

53. 55.

57. (a) (b)

The lines in (a) do not appear perpendicular, but they do in (b)

because a square setting is used. The lines are perpendicular.

59. (a) (b)

61. (a) (b)

63. (a) (b)

65. 67.

69. 71. Not collinear, because

73. 75.

77.

79. (a)

(b) (c) When six units are produced,

wages are $17.00 per hour

with either option. Choose

position 1 when less than

six units are produced and

position 2 otherwise.

81. (a)

(b) (c) 49 units

units

83. 85. 2 87. 89.

91. Proof 93. Proof 95. Proof 97. True


1. (a) Domain of Range of

Domain of Range of

(b)

(c) (d) (e) and

3. (a) (b) (c) (d)

5. (a) 3 (b) 0 (c) (d)

7. (a) 1 (b) 0 (c) 9.

11.

13. Domain: Range:

15. Domain: All real numbers such that where is an

integer; Range:

17. Domain: Range:

19. Domain:

21. Domain: All real numbers such that where is

an integer.

23. Domain:

25. (a) (b) 2 (c) 6 (d)

Domain: Range:

27. (a) 4 (b) 0 (c) (d)

Domain: Range:

29. 31.

Domain: Domain:

Range: Range:

33. 35.

Domain: Domain:

Range: Range:

37. The student travels during the first 4 min, is

stationary for the next 2 min, and travels during the

final 4 min.

39. is not a function of 41. is a function of

43. is not a function of 45. is not a function of

47. 48. 49. 50. 51. 52.

53. (a) (b)

x

−6

−2

−4

4

2

−2 2 4 6 8

y

x

−6

−2

−4

4

−4−6 −2 2 4

y

geacbd

x.yx.y

x.yx.y

1 miymin

12 miymin

t

y

2 3

−1

1

2

−1−2−3−4 1 2 3 4

−2

−3

1

2

4

5

x

y

f22, 2gf0, 3gs2`, `df23, 3g

gstd 5 2 sin p tf sxd 5 !9 2 x2

1 2 3

1

2

x

y

−2−4 2 4

2

4

6

8

x

y

f0, `ds2`, `df1, `ds2`, `d

hsxd 5 !x 2 1f sxd 5 4 2 x

s2`, 0g < f1, `ds2`, `d;2b222

s2`, 1d < f2, `ds2`, `d;2t 2 1 421

s2`, 23d < s23, `d

nx Þ 2np,x

f0, 1gs2`, 0d < s0, `ds2`, 0d < s0, `d;

s2`, 21g < f1, `dnt Þ 4n 1 2,t

s2`, 0gf23, `d;5 21yf!x 2 1s1 1 !x 2 1dg, x Þ 2

s!x 2 1 2 x 1 1dyfsx 2 2dsx 2 1dg3x2 1 3x Dx 1 sDxd2, Dx Þ 02

12

2 1 2t 2 t 221

2x 2 52b 2 32923

x < 2x < 21, x < 1,x < 1x 5 21

gs3d 5 24f s22d 5 21;

f24, 4gg:f23, 3g;g:

f23, 5gf :f24, 4g;f :

2!2s5!2dy212y 1 5x 2 169 5 0

xs655d 5 45

1500

0

0

50

x 5 s1330 2 pdy15

0

0

30

50

(6, 17)

W2 5 9.20 1 1.30xW1 5 12.50 1 0.75x;

728F < 22.28C5F 2 9C 2 160 5 0;

1b, a2 2 b2

c 210, 2a2 1 b2 1 c2

2c 2−1

−3 6(0, 0)

(2, 4)

m1 Þ m2y 5 2x

V 5 22000t 1 28,400V 5 125t 1 2040

y 2 5 5 0x 2 2 5 0

24x 1 40y 2 53 5 040x 2 24y 2 9 5 0

x 1 2y 2 4 5 02x 2 y 2 3 5 0

15

−10

−15

10

10

−10

−10

10

y

1

x32

1

2

3

12

x1

2

1

3

4

−2

−2

−3

−4

−3−4 2 3 4

y

Answers to Odd-Numbered Exercises A27



(c) (d)

(e) (f )

55. (a) Vertical translation (b) Reflection about the axis

(c) Horizontal translation

57. (a) 0 (b) 0 (c) (d)

(e) (f )

59. Domain:

Domain:

No, their domains are different.

61. Domain:

Domain:

No

63. (a) 4 (b)

(c) Undefined. The graph of does not exist at

(d) 3 (e) 2

(f ) Underfined. The graph of does not exist at


Example:

67. Even 69. Odd 71. (a) (b)

73. is even. is neither even nor odd. is odd.

75. 77.

79. 80. 81. 82.

83. (a)

(b) The changes in temperature will occur 1 hr later.

(c) The temperatures are lower.

85. (a) (b)

87.

89. Proof 91. Proof

93. (a) (b)

(c)

1

2

3

4

5

6

The dimensions of the box that yield a maximum volume

are

95. False. For example, if 97. True

99. Putnam Problem A1, 1988


1. Quadratic 3. Linear

5. (a) and (b) 7. (a)

(b)

(c) 3.63 cm

Approximately linear

(c) 136

9. (a)

(b) (c) Greater per capita electricity

consumption by a country

tends to relate to greater per

capita gross national product

of the country. Hong Kong,

Venezuela, South Korea

(d)

r < 0.968

y 5 0.134x 1 0.28

30

0

0

180

y = 0.124x + 0.82

r < 0.838

y 5 0.124x 1 0.82

0

10

0 110

d = 0.066F

x

y

3 6 9 12 15

50

100

150

200

250

d 5 0.066F

f sxd 5 x2, then f s21d 5 f s1d.4 3 16 3 16 cm.

6f24 2 2s6dg2 5 86424 2 2s6d 5f24 2 2s5dg2 5 98024 2 2s5d 4f24 2 2s4dg2 5 102424 2 2s4d 3f24 2 2s3dg2 5 97224 2 2s3d 2f24 2 2s2dg2 5 80024 2 2s2d 1f24 2 2s1dg2 5 48424 2 2s1d Volume, V Width Height, x

Length and

12

−100

−1

1100

4 3 16 3 16 cmVsxd 5 xs24 2 2xd2, x > 0

52x 2 2,

2,

22x 1 2,

if x ≥ 2

if 0 < x < 2

if x ≤ 0

f sxd 5 |x| 1 |x 2 2| 5

As15d < 345 acresyfarm

10 20 30 40 50

100

200

300

400

500

t

A

Aver

age

num

ber

of

acre

s per

far

m

Year (0 ↔ 1950)

18

Ts4d 5 168, Ts15d 5 248

iii, c 5 3iv, c 5 32i, c 514ii, c 5 22

y 5 2!2xf sxd 5 22x 2 5

hgf

s32, 24ds3

2, 4dhsxd 5 2xgsxd 5 x 2 2;f sxd 5 !x ;

x 5 24.f

x 5 25.g

22

s2`, 0d < s0, `dsg 8 f dsxd 5 9yx2 2 1;

s2`, 21d < s21, 1d < s1, ̀ ds f 8 gdsxd 53ysx2 2 1d;

s2`, `dsg 8 f dsxd 5 |x|;f0, `ds f 8 gdsxd 5 x;

x 2 1 sx ≥ 0d!x2 2 1

!1521

643

3

2

1

1

−1

−2

2

4

5x

y

4321

−1

−2

1

−3

x

y

4

3

2

1

4321

y

x

x-

x

−6

4

2

−4 −2 2 4 6

y

x

−2

−8

−10

−6

−4

−4 −2 4 6

y

x

−2

−8

−6

−4

−4 −2 2 4 6

y

x

−2

4

6

2

−4 −2 2 4 6

y





11. (a)

(b)

13. (a) Linear:

Cubic:

(b) (c) Cubic

(d)

(e) Linear:

million people

Cubic:

million people

(f ) Answers will vary.

15. (a)

(b) (c) 214

17. (a) Yes. At time there is one and only one displacement

(b) Amplitude: 0.35; Period: 0.5 (c)

(d) The model appears to fit

the data.


Review Exercises for Chapter P (page 37)

1. 3. 5. axis symmetry

7. 9.

11. 13.

15. 17. 19.

21. 23.

25. or 27. or

29. (a) (b)

(c) (d)

31.

33. Not a function 35. Function

37. (a) Undefined (b)

39. (a) Domain: Range:

(b) Domain: Range:

(c) Domain: Range: s2`, `ds2`, `d;s2`, 0d < s0, `ds2`, 5d < s5, `d;

f0, 6gf26, 6g;21ys1 1 Dxd , Dx Þ 0, 21

x

y

−1−2 3 4

−2

3

4

x

y

1 2 3 4 5 6−1

−2

−3

1

2

3

$9950V 5 12,500 2 850t;

x 1 2 5 02x 1 y 5 0

5x 2 3y 1 22 5 07x 2 16y 1 78 5 0

(−3, 0)

−4 −3 −1 1 2 3

−3

−4

1

2

3

x

y

(0, −5)

−3 −2 −1 1

1

−2

−3

−4

2 4x

y

2x 1 3y 1 6 5 03x 2 2y 2 10 5 0

y 5 223 x 2 2y 5

32x 2 5

x

y

1 2 3 4 5

1

2

3

4

5

, 132

52

5,( )

( )

t 573m 5

37

y 5 x3 2 4xs4, 1dXmin = -5

Xmax = 5

Xscl = 1

Ymin = -30

Ymax = 10

Yscl = 5

1 2 3 4 5

1

2

3

4

5

x

y

x

y

−5−10 5

5

10

−1−2−3 1

−1

2

3

x

y

−1 1 2 3

−1

−2

1

2

3

x

y

y-s1, 0d, s0, 12ds3

2, 0d, s0, 23d

0

0

4

0.9

(0.125, 2.35)

(0.375, 1.65)

y 5 0.35 sins4p td 1 2

y.t

0

0

7

300

y 5 21.806x3 1 14.58x2 1 16.4x 1 10

Ns14d < 51.5

Ns14d < 96.2

90

13

25

0

y 5 20.084t2 1 5.84t 1 26.7

90

13

25

0

y1

y2

y2 5 20.1289t3 1 2.235t2 2 4.86t 1 35.2

y1 5 4.83t 1 28.6

31.1 centsymi

8

0

0

y1

y2

y3

y1 + y

2 + y

3

15

y1 1 y2 1 y3 5 0.03434t 3 2 0.3451t 2 1 1.086t 1 8.47

y3 5 0.092t 1 0.79

y2 5 0.110t 1 2.07

y1 5 0.03434t 3 2 0.3451t 2 1 0.884t 1 5.61



41. (a) (b)

(c) (d)

43. (a)

All the graphs pass through the origin. The graphs of the

odd powers of are symmetric with respect to the origin

and the graphs of the even powers are symmetric with

respect to the axis. As the powers increase, the graphs

become flatter in the interval Graphs of these

equations with odd powers pass through Quadrants I and III.

Graphs of these equations with even powers pass through

Quadrants I and II.

(b) The graph of should pass through the origin and

Quadrants I and III. It should be symmetric with respect to

the origin and be fairly flat in the interval The

graph of should pass through the origin and

Quadrants I and II. It should be symmetric with respect to

the axis and be fairly flat in the interval

45. (a)

(b) Domain:

(c) Maximum area:

;

47. (a) Minimum degree: 3; Leading coefficient: negative

(b) Minimum degree: 4; Leading coefficient: positive

(c) Minimum degree: 2; Leading coefficient: negative

(d) Minimum degree: 5; Leading coefficient: positive

49. (a) Yes. For each time there corresponds one and only one

displacement

(b) Amplitude: 0.25; Period: 1.1 (c)

(d) The model appears to fit

the data.

P.S. Problem Solving (page 39)

1. (a) Center: Radius: 5

(b) (c) (d)

3.

(a) (b)

(c) (d)

(e) (f )

5. (a) Domain:

(b) Dimensions

yield maximum area of

(c)

7. Tsxd 5 f2!4 1 x2 1 !s3 2 xd2 1 1gy4

50 m 3 25 m; Area 5 1250 m2

1250 m2.

50 m 3 25 m

110

0

0

1600

s0, 100dAsxd 5 xfs100 2 xdy2g;

x1

1

3

4

−2 −1−1

−2

−3

−4

−3−4 2 3 4

y

x1

2

1

3

4

−2 −1−1

−2

−3

−4

−3−4 2 3 4

y

2Hsx 2 2d 1 2 5 51, x ≥ 2

2, x < 2

12Hsxd 5 5

12,

x ≥ 0

0, x < 0

x1

2

3

4

−2 −1−1

−2

−3

−4

−3−4 2 3 4

y

x1

2

1

3

4

−2 −1−1

−2

−3

−4

−3−4 2 3 4

y

Hs2xd 5 51, x ≤ 0

0, x > 02Hsxd 5 521, x ≥ 0

0, x < 0

x1

2

1

3

4

−2 −1−1

−2

−3

−4

−3−4 2 3 4

y

x1

2

1

3

4

−2 −1−1

−3

−4

−3−4 2 3 4

y

Hsx 2 2d 5 51, x ≥ 2

0, x < 2Hsxd 2 2 5 521, x ≥ 0

22, x < 0

x1

2

1

3

4

−2 −1−1

−2

−3

−4

−3−4 2 3 4

y

s3, 294dy 5

34 x 2

92y 5 2

34 x

s3, 4d;

0 2.2

−0.5

0.5

(0.5, −0.25)

(1.1, 0.25)

y 514 coss5.7td

y.

t

6 3 6 in.36 in.2

0

0

12

40

s0, 12dA 5 xs12 2 xd

s21, 1d.y-

y 5 x8

s21, 1d.

y 5 x7

21 < x < 1.

y-

x

3

0

−3

h

g

f

4

3

−2

−3

f

h

g2

3

3

2

1

21123x

y

0

2

c

c

2c

42

1

1

2

2

x

y

2c

0c

c 2

3

1

22

2

3

x

y0c

2c

2c

3

3

1

223x

y

2

c

c

0

c 2





9. (a) 5, less (b) 3, greater (c) 4.1, less

(d) (e) 4; Answers will vary.

11. (a) 13. Answers will vary.

(b)

−2−4−8 2 4−2

−6

2

6

8

x

y

sx 1 3d21 y2

5 18

(− 2 , 0) ( 2 , 0)

(0, 0)

x

y

−2

−2

−1

1

2

2

x 5 23 2 !18 < 27.2426

x 5 23 1 !18 < 1.2426,

4 1 h


This page contains answers for this chapter only

This page contains answers for this chapter only

section p.3 functions and their graphs functions …€¦ · section p.3 functions and their graphs...

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