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
Function name is independent variable is
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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SECTION P.3 Functions and Their Graphs 21
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
22 CHAPTER P Preparation for Calculus
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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SECTION P.3 Functions and Their Graphs 23
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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24 CHAPTER P Preparation for Calculus
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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SECTION P.3 Functions and Their Graphs 25
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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26 CHAPTER P Preparation for Calculus
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
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! " # $ % # & ' ( ) ! * + , - ' # . % / % # .
SECTION P.3 Functions and Their Graphs 27
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
Copyright 2010 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.
! " # $ % # & ' ( ) ! * + , - ' # . % / % # .
28 CHAPTER P Preparation for Calculus
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.
Copyright 2010 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.
! " # $ % # & ' ( ) ! * + , - ' # . % / % # .
SECTION P.3 Functions and Their Graphs 29
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
Copyright 2010 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.
! " # $ % # & ' ( ) ! * + , - ' # . % / % # .
30 CHAPTER P Preparation for Calculus
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
Copyright 2010 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.
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
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0 1 2 3 4 5 3 6 7 8 9 1 : ; < = 7 3 > 5 ? 5 3 >
32 CHAPTER P Preparation for Calculus
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
Copyright 2010 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.
0 1 2 3 4 5 3 6 7 8 9 1 : ; < = 7 3 > 5 ? 5 3 >
SECTION P.4 Fitting Models to Data 33
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.
Copyright 2010 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.
0 1 2 3 4 5 3 6 7 8 9 1 : ; < = 7 3 > 5 ? 5 3 >
34 CHAPTER P Preparation for Calculus
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.
(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 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)
(a) Use the regression capabilities of a graphing utility to find a
linear model for the data. What is the correlation coefficient?
(b) Use a graphing utility to plot the data and graph the model.
(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)
Copyright 2010 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.
0 1 2 3 4 5 3 6 7 8 9 1 : ; < = 7 3 > 5 ? 5 3 >
SECTION P.4 Fitting Models to Data 35
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.
(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 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) Use the regression capabilities of a graphing utility to find
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.
(b) Use a graphing utility to plot the data and graph the model.
(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) Use the regression capabilities of a graphing utility to find
a quadratic model for the data.
(b) Use a graphing utility to plot the data and graph the model.
(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
Copyright 2010 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.
0 1 2 3 4 5 3 6 7 8 9 1 : ; < = 7 3 > 5 ? 5 3 >
36 CHAPTER P Preparation for Calculus
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) Use the regression capabilities of a graphing utility to find
a cubic model for the data.
(b) Use a graphing utility to plot the data and graph the model.
(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) Use the regression capabilities of a graphing utility to find
a cubic model for the data.
(b) Use a graphing utility to plot the data and graph the model.
(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
Copyright 2010 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.
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.
Copyright 2010 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.
@ A B C D E C F G H I A J K L M G C N E O E C N
38 CHAPTER P Preparation for Calculus
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) 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.
(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
Copyright 2010 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.
@ 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?
Figure for 1 Figure for 2
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.
Figure for 5 Figure for 6
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
Copyright 2010 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.
P Q R S T U S V W X Y Q Z [ \ ] W S ^ U _ U S ^
40 CHAPTER P Preparation for Calculus
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
slope of the tangent line at greater than or less than
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
slope of the tangent line at greater than or less than
this number?
(b) Find the slope of the line joining and . Is the
slope of the tangent line at greater than or less than
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.
Figure for 11 Figure for 12
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
Copyright 2010 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.
P Q R S T U S V W X Y Q Z [ \ ] W S ^ U _ U S ^
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.
11. Answers will vary.
13. 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
Copyright 2010 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.
P Q R S T U S V W X Y Q Z [ \ ] W S ^ U _ U S ^
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.
Section P.2 (page 16)
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
Copyright 2010 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.
` 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
Section P.3 (page 27)
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
Copyright 2010 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.
` a b c d e c f g h i a j k l m g c n e o e c n
(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
65. Answers will vary.
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
Section P.4 (page 34)
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
A28 Answers to Odd-Numbered Exercises
Copyright 2010 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.
` a b c d e c f g h i a j k l m g c n e o e c n
Answers to Odd-Numbered Exercises A29
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.
19. Answers will vary.
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
Copyright 2010 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.
` a b c d e c f g h i a j k l m g c n e o e c n
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
A30 Answers to Odd-Numbered Exercises
Copyright 2010 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.
` a b c d e c f g h i a j k l m g c n e o e c n
Answers to Odd-Numbered Exercises A31
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
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This page contains answers for this chapter only
This page contains answers for this chapter only