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c03a
...
The original price of the shirt was $100,
but it has been discounted 30%
...
How much will you pay for the shirt?
Naïve shoppers might be lured into thinking this shirt will cost $50 because they add the 20% and 30% to
get 50% off, but they will end up paying more than that
...
Experienced shoppers have already learned composition of
functions
...
One function takes an input
(original price, $100) and maps it to an output (sale price, $70), and then another function takes that output
as its input (sale price, $70) and maps that to an output (checkout price, $56)
...
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1/18/12
1:11 PM
Page 267
I N T H I S C H A P T E R you will find that functions are part of our everyday thinking: converting from degrees Celsius to
degrees Fahrenheit, DNA testing in forensic science, determining stock values, and the sale price of a shirt
...
First, we will establish what a relation is, and then we will determine
whether a relation is a function
...
We
will determine whether a function is increasing or decreasing on an interval and calculate the average rate of change of a
function
...
We will discuss one-to-one functions and inverse
functions
...
F U N CTI O N S AN D TH E I R G R AP H S
3
...
2
3
...
4
3
...
6
Functions
Graphs of
Functions;
PiecewiseDefined
Functions;
Increasing
and
Decreasing
Functions;
Average Rate
of Change
Graphing
Techniques:
Operations
on Functions
and
Composition
of Functions
One-to-One
Functions
and Inverse
Functions
Modeling
Functions
Using
Variation
• Adding,
Subtracting,
Multiplying,
and Dividing
Functions
• Composition
of Functions
• Determine
Whether
a Function Is
One-to-One
• Inverse
Functions
• Graphical
Interpretation
of Inverse
Functions
• Finding the
Inverse
Function
• Direct
Variation
• Inverse
Variation
• Joint
Variation and
Combined
Variation
• Relations
and
Functions
• Functions
Defined by
Equations
• Function
Notation
• Domain of a
Function
• Recognizing
and
Classifying
Functions
• Increasing
and
Decreasing
Functions
• Average Rate
of Change
• PiecewiseDefined
Functions
Transformations
• Horizontal
and Vertical
Shifts
• Reflection
about the
Axes
• Stretching
and
Compressing
LEARNING OBJECTIVES
■
■
■
■
■
■
Find the domain and range of a function
...
Sketch graphs of general functions employing translations of common functions
...
Find the inverse of a function
...
267
c03a
...
1
F U N CTI O N S
C O N C E P TUAL O BJ E CTIVE S
S K I LLS O BJ E CTIVE S
■
■
■
■
■
Determine whether a relation is a function
...
Use function notation
...
Determine the domain and range of a function
...
Understand that all functions are relations but not all
relations are functions
...
Temperature is some specific value at a particular time of day
...
First-class postage rates correspond to the weight of a letter
...
They all describe a particular correspondence between two groups
...
The first set is called the domain, and the corresponding
second set is called the range
...
DEFINITION
Relation
A relation is a correspondence between two sets where each element in the first set,
called the domain, corresponds to at least one element in the second set, called the
range
...
The domain is the set of all the first
components of the ordered pairs, and
the range is the set of all the second
components of the ordered pairs
...
{Michael, Tania, Dylan, Trevor, Megan}
The range is the set of all the
second components
...
268
c03a
...
1 Functions
DEFINITION
269
Function
A function is a correspondence between two sets where each element in the first set,
called the domain, corresponds to exactly one element in the second set, called the
range
...
For a relation, each input corresponds to at least one output, whereas, for a function, each
input corresponds to exactly one output
...
Also note that the range (set of values to which the elements of the domain correspond)
is a subset of the set of all blood types
...
For example, at a university, four primary sports TIME OF DAY
C OMPETITION
typically overlap in the late fall: football, volleyball, soccer,
Football
and basketball
...
M
...
2:00 P
...
Volleyball
7:00 P
...
Michael
Megan
Dylan
Trevor
Tania
A
AB
O
B
Basketball
W OR DS
M ATH
The 1:00 start time corresponds
to exactly one event, Football
...
The 7:00 start time corresponds
to two events, Soccer and Basketball
...
M
...
a
...
{(Ϫ3, 4), (2, 4), (3, 5), (2, 2)}
c
...
No x-value is repeated
...
This relation is a function
...
The value x ϭ 2 corresponds to both y ϭ 2 and y ϭ 4
...
ATHLETIC
EVENT
1:00 P
...
2:00 P
...
(7:00 P
...
, Soccer)
(7:00 P
...
, Basketball)
Determining Whether a Relation Is a Function
Range
Not a
Function
START
TIME
(2:00 P
...
, Volleyball)
Because an element in the domain, 7:00 P
...
, corresponds to more than one element in the
range, Soccer and Basketball, this is not a function
...
EXAMPLE 1
BLOOD
TYPE
Soccer
7:00 P
...
PEOPLE
Football
Volleyball
Soccer
Basketball
7:00 P
...
Study Tip
All functions are relations but not all
relations are functions
...
1
...
a
...
{(Ϫ1, Ϫ2), (Ϫ2, Ϫ3),
(Ϫ3, Ϫ4), (Ϫ4, Ϫ5)}
c
...
no b
...
yes
c
...
This relation is a function
...
a
...
{(1, 2), (1, 3), (5, 6), (7, 8)}
c
...
M
...
M
...
M
...
function
b
...
function
c03a
...
A function can also be defined algebraically by
an equation
...
This equation
assigns to each x-value exactly one corresponding y-value
...
2
Not all equations are functions
...
2)2 Ϫ 3(1
...
However, if we reverse the
independent and dependent variables,
then x ϭ y2 is a function of y
...
16
Since the variable y depends on what value of x is selected, we denote y as the dependent
variable
...
Although functions are defined by equations, it is important to recognize that not all
equations are functions
...
Throughout the
ensuing discussion, we assume x to be the independent variable and y to be the dependent
variable
...
Some ordered pairs that correspond to these functions are
y ϭ x2:
yϭƒxƒ:
y ϭ x3:
(Ϫ1, 1) (0, 0) (1, 1)
(Ϫ1, 1) (0, 0) (1, 1)
(Ϫ1, Ϫ1) (0, 0) (1, 1)
The fact that x ϭ Ϫ1 and x ϭ 1 both correspond to y ϭ 1 in the first two examples does not
violate the definition of a function
...
Some ordered pairs that correspond to these equations are
x2 ϩ y2 ϭ 1
y = ; 21 - x2
xϭƒyƒ
y = ;x
R ELATION
x ϭ y2
S OLVE R ELATION
y = ;x
FOR Y
P OINTS THAT LIE
ON THE
G RAPH
(1, Ϫ1) (0, 0) (1, 1)
x ϭ 1 maps to both y ϭ Ϫ1 and y ϭ 1
(0, Ϫ1) (0, 1) (Ϫ1, 0) (1, 0)
x ϭ 0 maps to both y ϭ Ϫ1 and y ϭ 1
(1, Ϫ1) (0, 0) (1, 1)
x ϭ 1 maps to both y ϭ Ϫ1 and y ϭ 1
c03a
...
1 Functions
271
Let’s look at the graphs of the three functions of x:
y
y
y
x
x
x
y = |x|
y = x2
y = x3
Let’s take any value for x, say x ϭ a
...
A function of x maps each x-value to exactly one y-value; therefore, there should be at
most one point of intersection with any vertical line
...
Look at the graphs of the three
equations that do not represent functions of x
...
Thus, there are two y-values that correspond to
some x-value in the domain, which is why these equations do not define y as a function of x
...
This test
is called the vertical line test
...
c03a
...
1
...
a
...
a
...
y
y
x
x
Solution:
b
...
a
...
y
y
x
Answer: a
...
no
Classroom Example 3
...
2*
Let a be a positive real number
...
Because the vertical line intersects the graph of the equation at two points, this equation
does not represent a function
...
Because any vertical line will intersect the graph of this equation at no more than one
point, this equation represents a function
...
■
Answer: The graph of the equation
is a circle, which does not pass the
vertical line test
...
■ YO U R T U R N
Determine whether the equation (x Ϫ 3)2 ϩ (y ϩ 2)2 ϭ 16 is a
function of x
...
This is sometimes called the Rule of 4
...
N UMERICALLY
ALGEBRAICALLY
G RAPHICALLY
y
{(Ϫ3, 3), (Ϫ1, 1), (0, 0), (1, 1), (5, 5)}
y ϭ ƒxƒ
x
c03a
...
1 Functions
273
Function Notation
We know that the equation y ϭ 2x ϩ 5 defines y as a function of x because its graph is a
nonvertical line and thus passes the vertical line test
...
The output is found by taking 2 times
the input and then adding 5
...
In other words, y ϭ ƒ(x)
...
In other words, the function ƒ maps some value x in the domain to some value f(x) in the
range
...
To evaluate
functions, it is often useful to think of the independent variable or argument as a placeholder
...
” Any expression can be substituted for the argument:
f(1) = (1)2 - 3(1)
f(x + 1) = (x + 1)2 - 3(x + 1)
f(-x) = (-x)2 - 3(-x)
It is important to note:
■
■
■
■
ƒ(x) does not mean f times x
...
Other common function names are g and G, but any letter can be used
...
The letter t is also
common because in real-world applications it represents time, but any letter can be
used
...
Study Tip
It is important to note that ƒ(x) does
not mean ƒ times x
...
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C H A P T E R 3 Functions and Their Graphs
EXAMPLE 3
Classroom Example 3
...
3
Let f(x) ϭ Ϫx4 Ϫ x Ϫ 1
...
f(Ϫ1)
b
...
Solution:
ƒ(Ϫ1) ϭ 2(Ϫ1)3 Ϫ 3(Ϫ1)2 ϩ 6
Evaluate the right side
...
Classroom Example 3
...
4
Consider this graph:
ƒ( ) ϭ 2( )3 Ϫ 3( )2 ϩ 6
To find ƒ(Ϫ1), substitute x ϭ Ϫ1 into
the function
...
Ϫ1 b
...
ƒ(Ϫ1) ϭ 1
EXAMPLE 4
Finding Function Values from the Graph
of a Function
The graph of ƒ is given on the right
...
b
...
d
...
f
...
Find ƒ(1)
...
Find 4ƒ(3)
...
Find x such that ƒ(x) ϭ 2
...
Find f(Ϫ3)
...
Find f(2)
...
* Find x such that f(x) ϭ 0
...
ƒ(0) ϭ 5
Answer:
a
...
Ϫ1
Solution (c): The value x ϭ 2 corresponds to the value y ϭ 1
...
1
...
ƒ(1) ϭ 2
Solution (d): The value x ϭ 3 corresponds to the value y ϭ 2
...
2 ϭ 8
Solution (e): The value y ϭ 10 corresponds to the value x ϭ 5
...
ƒ(Ϫ1) ϭ 2
b
...
3ƒ(2) ϭ Ϫ21
d
...
For the following graph of a function, find:
a
...
ƒ(0)
c
...
the value of x that corresponds to ƒ(x) ϭ 0
y
(–2, 9)
10
(0, 1)
x
(–1, 2)
(1, 0)
–5
(2, –7)
–10
5
c03a
...
1 Functions
EXAMPLE 5
Evaluating Functions with Variable
Arguments (Inputs)
For the given function ƒ(x) ϭ x2 Ϫ 3x, evaluate ƒ(x ϩ 1) and simplify if possible
...
★
275
Classroom Example 3
...
5
Let f(x) ϭ 1 Ϫ (x Ϫ3)2
...
f(x ϩ 3)
b
...
f(2x ϩ 1)
Answer:
a
...
1 Ϫ x2
2
c
...
Write the original function
...
Z x2 - 3x - 2
f ( ) ϭ ( ) Ϫ 3( )
2
Substitute x ϩ 1 for the argument
...
f (x + 1) = x2 + 2x + 1 - 3x - 3
Combine like terms
...
Evaluating Functions: Sums
For the given function H(x) ϭ x ϩ 2x, evaluate:
2
a
...
H(x) ϩ H(1)
Classroom Example 3
...
6
Let f(x) ϭ 2 Ϫ x2
...
f(x ϩ 1)
b
...
Ϫx2 Ϫ 2x ϩ 1
b
...
■
Answer: g(x - 1) = x2 - 4x + 6
Technology Tip
Use a graphing utility to display
graphs of
y1 ϭ H(x ϩ 1) ϭ (x ϩ 1)2 ϩ 2(x ϩ 1)
and y2 ϭ H(x) ϩ H(1) ϭ x2 ϩ 2x ϩ 3
...
2
H(x ϩ 1) ϭ (x ϩ 1) ϩ 2(x ϩ 1)
Eliminate the parentheses on the right side
...
H(x + 1) = x2 + 4x + 3
Solution (b):
Write H(x)
...
H(1) 3 ؍ )1(2 ؉ 2)1( ؍
Evaluate the sum H(x) ϩ H(1)
...
The graphs are not the same
...
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276
7:34 PM
Page 276
C H A P T E R 3 Functions and Their Graphs
EXAMPLE 7
Technology Tip
Use a graphing utility to display
graphs of y1 ϭ G(Ϫx) ϭ (Ϫx)2 Ϫ (Ϫx)
and y2 ϭ ϪG(x) ϭ Ϫ(x2 Ϫ x)
...
ϪG(t)
a
...
G( ) ϭ ( )2 Ϫ ( )
Substitute Ϫt for the argument of G
...
G(- t) = t2 + t
Solution (b):
The graphs are not the same
...
Multiply by Ϫ1
...
- G(t) = - t2 + t
Note: Comparing the results of part (a) and part (b), we see that G(؊ t)
EXAMPLE 8
Classroom Example 3
...
7*
Let f(x) ϭ Ϫ3 Ϫ x Ϫ 2x2
...
Answer: 3 Ϫ x ϩ 2x2
Classroom Example 3
...
8
Let f(x) ϭ 1 Ϫ (x Ϫ 1)2
...
b
...
0
▼
b
...
F a b
b
...
Replace the argument with 1
...
1
13
Fa b =
2
2
Solution (b):
Evaluate F(1)
...
F(2) ϭ 3(2) ϩ 5 ϭ 11
Divide F(1) by F(2)
...
G(t Ϫ 2) ϭ 3t Ϫ10
b
...
ϭG(3)
5
1
d
...
■ YO U R T U R N
F(1)
...
G(t Ϫ 2)
b
...
G(1)
G(3)
1
d
...
1 Functions
277
Now that we have shown that f(x + h) Z f(x) + f(h), we turn our attention to one of
the fundamental expressions in calculus: the difference quotient
...
2
...
In Section 3
...
EXAMPLE 9
Evaluating the Difference Quotient
For the function f (x) = x2 - x, find
Classroom Example 3
...
9
Let f (x) ϭ Ϫ1 Ϫ x Ϫ 2x 2
...
f(x + h) - f (x)
, h Z 0
...
ƒ( ) ϭ ( )2 Ϫ ( )
Calculate ƒ(x ϩ h)
...
Answer: -(1 + 2h + 4x)
f(x + h) - f(x)
h
2
Let ƒ(x ϩ h) ϭ (x ϩ h)2 Ϫ (x ϩ h) and ƒ(x) ϭ x2 Ϫ x
...
qxd
h Z 0
Eliminate the parentheses inside the
first set of brackets
...
=
x2 + 2xh + h2 - x - h - x2 + x
h
Combine like terms
...
=
h(2x + h - 1)
h
Divide out the common factor, h
...
■
Answer: 2x + h
Domain of a Function
Sometimes the domain of a function is stated explicitly
...
Every negative
real number in the domain is mapped to a positive real number in the range through the
absolute value function
...
qxd
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C H A P T E R 3 Functions and Their Graphs
f(x) = 1x
If the expression that defines the function is given but the domain is not stated explicitly,
then the domain is implied
...
For example,
does not have the domain explicitly stated
...
Note that
if the argument is negative, that is, if x Ͻ 0, then the result is an imaginary number
...
f (x) = 1x
F UNCTION
I MPLICIT D OMAIN
[0, ϱ)
In general, we ask the question, “what can x be?” The implicit domain of a function excludes
values that cause a function to be undefined or have outputs that are not real numbers
...
Graph of F(x) = 2
is shown
...
H(x) = 19 - 2x
State the domain of the given functions
...
F(x) =
3
x - 25
2
3
c
...
F(x) =
Determine any restrictions on the
values of x
...
State the domain restrictions
...
x2 Z 25 or x Z ; 125 = ;5
x Z ;5
Write the domain in interval notation
...
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11/24/11
6:02 PM
Page 279
Write the original equation
...
9 Ϫ 2x Ն 0
Solve the restriction inequality
...
1 Functions
Classroom Example 3
...
10
Find the domain of these
functions
...
f (x) = 2
2x - 32
6
b
...
* f (x) = 3
A x
4
State the domain restrictions
...
3
G(x) = 1x - 1
Determine any restrictions on the
values of x
...
R
Write the domain in interval notation
...
Solution (c):
■ YO U R T U R N
a
...
* f (x) =
Answer:
a
...
(- ϱ, - 3]
c
...
(- ϱ, ϱ)
■
State the domain of the given functions
...
g(x) =
1
x2 - 4
Applications
Functions that are used in applications often have restrictions on the domains due to
physical constraints
...
The function ƒ(x) ϭ x3 has no restrictions on x, and therefore
the domain is the set of all real numbers
...
EXAMPLE 11
Price of Gasoline
Following the capture of Saddam Hussein in Iraq in 2003, gas prices in the United
States escalated and then finally returned to their precapture prices
...
05x2 ϩ 0
...
7, where C is the cost function and x represents the number
of months after the capture
...
Determine the domain of the cost function
...
What was the average price of gas per gallon 3 months after the capture?
Solution (a):
Since the cost function C(x) ϭ Ϫ0
...
3x ϩ 1
...
Solution (b):
Write the cost function
...
05x2 ϩ 0
...
7
Find the value of the function
when x ϭ 3
...
05(3)2 ϩ 0
...
7
Simplify
...
15
The average price per gallon 3 months after the capture was $2
...
279
0ՅxՅ6
Answer: a
...
x Z ; 2 or
( - ϱ, - 2) ഫ ( -2, 2) ഫ(2, ϱ)
c03a
...
Solution:
The volume of any rectangular box is V ϭ lwh, where V is the volume, l is the length, w is
the width, and h is the height
...
Write the volume as a function of depth d
...
V(d ) = 300d
Determine any restrictions on the domain
...
1
S U M MARY
Relations and Functions (Let x represent the independent variable and y the dependent variable
...
Relation
x
x ϭ y2
y
Every x-value in the domain maps
to exactly one y-value in the range
...
Functions can be represented by equations
...
I NPUT
CORRESPONDENCE
OUTPUT
EQUATION
x
Function
y
y ϭ 2x ϩ 5
Independent
variable
Mapping
Dependent
variable
Mathematical
rule
Argument
ƒ
ƒ(x)
ƒ(x) ϭ 2x ϩ 5
The domain is the set of all inputs (x-values), and the range is the set of all corresponding outputs (y-values)
...
f (x) = 3x2 + 2x
f ( ) ϭ 3( )2 ϩ 2( )
Explicit domain is stated, whereas implicit domain is found by excluding x-values that:
■
■
make the function undefined (denominator ϭ 0)
...
c03a
...
1 Functions
281
SECTION
3
...
Assume that the coordinate pair (x, y) represents the
independent variable x and the dependent variable y
...
2
...
Domain
Range
Domain
Range
Domain
MONTH
AVERAGE
TEMPERATURE
PERSON
10-DIGIT PHONE #
START TIME
NFL GAME
1:00 P
...
4:00 P
...
7:00 P
...
• Bucs/Panthers
• Bears/Lions
• Falcons/Saints
• Rams/Seahawks
• Packers/Vikings
October
January
April
Mary
Jason
Chester
78°F
68°F
4
...
Domain
Chris
Alex
Morgan
Range
Domain
Range
PERSON
Carrie
Michael
Jennifer
Sean
DATE THIS
WEEKEND
Jordan
Pat
6
...
{(0, Ϫ3), (0, 3), (Ϫ3, 0), (3, 0)}
11
...
x ϩ y ϭ 9
14
...
10
...
{(0, 1), (1, 1), (2, 1), (3, 1)}
15
...
y
A
B
8
...
{(0, 0), (9, Ϫ3), (4, Ϫ2), (4, 2), (9, 3)}
2
Range
16
...
y
17
...
y ϭ 3
y
(0, 5)
x
(–5, 0)
x
x
(5, 0)
(0, –5)
22
...
y
x
24
...
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Page 282
C H A P T E R 3 Functions and Their Graphs
In Exercises 25–32, use the given graphs to evaluate the functions
...
y ϭ ƒ(x)
26
...
y ϭ p(x)
y
y
y
5
5
(2, 5)
y
7
(–5, 7)
(1, 3)
10
(–3, 5)
(1, 5)
(–1, 4)
(–3, 1)
(0, 1)
–5
28
...
y ϭ C(x)
30
...
r(Ϫ4) b
...
r (3)
31
...
y ϭ T(x)
y
y
4
5
–5
(7, –3)
(3, –5)
–10
a
...
p(0) c
...
g(Ϫ3) b
...
g(5)
a
...
ƒ(0) c
...
C(2) b
...
C(Ϫ2)
–5
–8
a
...
q(0) c
...
S(Ϫ3) b
...
S(2)
33
...
34
...
35
...
36
...
37
...
38
...
39
...
a
...
T(Ϫ2) c
...
Find x if T(x) ϭ 4 in Exercise 32
...
ƒ(x) ϭ 2x Ϫ 3
F(t) ϭ 4 Ϫ t 2
g(t) ϭ 5 ϩ t
G(x) ϭ x2 ϩ 2x Ϫ 7
41
...
G(Ϫ3)
43
...
F(Ϫ1)
45
...
G(Ϫ3) Ϫ F(Ϫ1)
47
...
2F(Ϫ1) Ϫ 2G(Ϫ3)
49
...
ƒ(x ϩ 1) Ϫ ƒ(x Ϫ 1)
50
...
F(t ϩ 1) Ϫ F(t Ϫ 1)
51
...
g(x ϩ a) Ϫ ƒ(x ϩ a)
52
...
G(x ϩ b) ϩ F(b)
In Exercises 57–64, evaluate the difference quotients using the same ƒ, F, G, and g given for Exercises 41–56
...
f (x + h) - f (x)
h
58
...
g(t + h) - g(t)
h
60
...
f (- 2 + h) - f (-2)
h
62
...
g(1 + h) - g(1)
h
64
...
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Page 283
3
...
Express the domain in interval notation
...
ƒ(x) ϭ 2x Ϫ 5
69
...
f(x) ϭ Ϫ2x Ϫ 5
x + 5
x - 5
70
...
G(t) =
77
...
F(x) =
81
...
P(x) =
89
...
G(x) =
2t - t - 6
5
t
2
Ϫ1/2
93
...
Q(x) =
90
...
T(x) =
2
x - 4
83
...
R(x) =
t - 3
76
...
R(x) =
2
79
...
h(x) ϭ 3x4 Ϫ 1
75
...
g(x) = 15 - 2x
1
1x - 3
2
67
...
F(x) = 2x2 - 25
5
84
...
p(x) =
225 - x2
x2
91
...
p(x) ϭ (x Ϫ 1)2 (x2 Ϫ 9)
92
...
f (x) = 2x 5
96
...
Let g(x) ϭ x2 Ϫ 2x Ϫ 5 and find the values of x that correspond to g(x) ϭ 3
...
Let g(x) = 5 x 6
3
4
and find the value of x that corresponds to g(x) = 2
...
Let ƒ(x) ϭ 2x(x Ϫ 5)3 Ϫ 12(x Ϫ 5)2 and find the values of x that correspond to ƒ(x) ϭ 0
...
Let ƒ(x) ϭ 3x(x ϩ 3)2 Ϫ 6(x ϩ 3)3 and find the values of x that correspond to ƒ(x) ϭ 0
...
Budget: Event Planning
...
Write the cost of the reception in terms of
the number of guests and state any domain restrictions
...
Budget: Long-Distance Calling
...
10 per minute
for any domestic long-distance calls
...
103
...
The average temperature in Tampa,
Florida, in the springtime is given by the function
T(x) ϭ Ϫ0
...
8x Ϫ 10
...
What is
the temperature at 6 A
...
? What is the temperature at noon?
104
...
A firecracker is launched
straight up, and its height is a function of time,
h(t) ϭ Ϫ16t 2 ϩ 128t, where h is the height in feet and t is
the time in seconds with t ϭ 0 corresponding to the instant
it launches
...
Collectibles
...
When
Rodriguez was traded from the Texas Rangers to the New
York Yankees in 2004, the going rate for a signed baseball
card on eBay was P(x) = 10 + 1400,000 - 100x, where
x represents the number of signed cards for sale
...
qxd
284
11/25/11
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Page 284
C H A P T E R 3 Functions and Their Graphs
106
...
In Exercise 105, what was the lowest price on
eBay, and how many cards were available then? What was
the highest price on eBay, and how many cards were
available then?
107
...
An open box is constructed from a square 10-inch
piece of cardboard by cutting squares of length x inches out
of each corner and folding the sides up
...
108
...
A cylindrical water basin will be built to harvest
rainwater
...
Write a function representing the volume
of water V as a function of height h
...
48 gallons
...
S
...
Assume the exchange rate E(t) is a function of time (week);
let E(1) be the exchange rate during Week 1
...
S
...
Environment: Tossing the Envelopes
...
Suppose each of these adults throws away a
dozen envelopes per week
...
The width of the window of an envelope is 3
...
Create the function A(x) that
represents the area of the window in square inches
...
b
...
5) and explain what this value represents
...
Assume the dimensions of the envelope are 8 inches by
4 inches
...
5)
...
114
...
Each month, Jack
receives his bank statement in a 9
...
Each month, he throws away the envelope after
removing the statement
...
The width of the window of the envelope is 2
...
Create the function A(x) that
represents the area of the window in square inches
...
b
...
25) and explain what this value represents
...
Evaluate A(10)
...
Refer to the table below for Exercises 115 and 116
...
YEAR
1 2 3 4 5 6 7 8 9 10
Week
109
...
Approximate the exchange rates of the U
...
dollar to the nearest yen during Weeks 4, 7, and 8
...
Economics
...
S
...
For Exercises 111–112, refer to the following:
An epidemiological study of the spread of malaria in a rural area
finds that the total number P of people who contracted malaria t
days into an outbreak is modeled by the function
1
P(t) = - t 2 + 7t + 180
4
1 … t … 14
111
...
How many people have contracted
malaria 14 days into the outbreak?
112
...
How many people have contracted
malaria 6 days into the outbreak?
FED
...
45
2001
5
...
73
2003
1
...
00
2005
2
...
50
2007
5
...
50
115
...
Is the relation whose domain is the year and whose
range is the average federal funds rate for the month of
January a function? Explain
...
Finance
...
11/24/11
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3
...
Note: The following years were interpolated: 1989–1992;
1994–1995; 1997–1998
...
naftc
...
edu
Let the functions f, F, g, G, and H represent the number of tons of
carbon emitted per year as a function of year corresponding to
cement production, natural gas, coal, petroleum, and the total
amount, respectively
...
TOTAL H EALTH CARE COST
FOR FAMILY PLANS
1989
1993
1997
119
...
Estimate (to the
nearest thousand) the value of
a
...
g(50)
c
...
Environment: Global Climate Change
...
Write the five ordered pairs resulting from the table
...
Health Care Costs
...
Round
dollars to the nearest $1000
...
Health Care Costs
...
Is this relation a function? Explain
...
qxd
C AT C H T H E M I S TA K E
In Exercises 121–126, explain the mistake that is made
...
Determine whether the
relationship is a function
...
Given the function H(x) ϭ 3x Ϫ 2, evaluate the quantity
H(3) Ϫ H(Ϫ1)
...
What mistake was made?
x
Solution:
Solution: ƒ(x ϩ 1) ϭ ƒ(x) ϩ ƒ(1) ϭ x2 Ϫ x ϩ 0
ƒ(x ϩ 1) ϭ x2 Ϫ x
y
This is incorrect
...
Determine the domain of the function g(t) = 13 - t and
express it in interval notation
...
What mistake was made?
Apply the
horizontal line test
...
123
...
x
Solution:
What can t be? Any nonnegative real number
...
What mistake was made?
c03a
...
Given the function G(x) ϭ x2, evaluate
G(-1 + h) - G(-1)
...
Given the functions ƒ(x) ϭ ƒ x Ϫ A ƒ Ϫ 1 and ƒ(1) ϭ Ϫ1, find A
...
Ϫ1 ϭ ƒ Ϫ1 Ϫ A ƒ Ϫ1
Add 1 to both sides of the equation
...
This is incorrect
...
What mistake was made?
■
CONCEPTUAL
In Exercises 127–130, determine whether each statement is true or false
...
If a vertical line does not intersect the graph of an equation,
then that equation does not represent a function
...
If a horizontal line intersects a graph of an equation more
than once, the equation does not represent a function
...
If ƒ(Ϫa) ϭ ƒ(a), then ƒ may or may not represent a function
...
If ƒ(x) ϭ Ax2 Ϫ 3x and ƒ(1) ϭ Ϫ1, find A
...
If g(x) =
129
...
■
CHALLENGE
133
...
C and D
...
Construct a function that is undefined at x ϭ 5 and whose
graph passes through the point (1, Ϫ1)
...
b - x
135
...
f (x) = -52x2 - a2
TECH NOLOGY
137
...
What time of day is it the warmest? What is
the temperature? Looking at this function, explain why this
model for Tampa, Florida, is valid only from sunrise to
sunset (6 to 18)
...
Using a graphing utility, graph the height of the firecracker
in Exercise 104
...
139
...
What are the lowest and highest prices of the cards?
Does this agree with what you found in Exercise 106?
140
...
The thin chocolate coating on
a malted milk ball can be approximated by the surface area,
S(r) ϭ 4 r 2
...
Let f (x) = x2 + 1
...
Describe how the graph of y2 can
be obtained from the graph of y1
...
Let f (x) = 4 - x2
...
Describe how the graph of y2 can
be obtained from the graph of y1
...
qxd
11/24/11
4:53 PM
SECTION
3
...
Determine whether functions are increasing,
decreasing, or constant
...
Evaluate the difference quotient for a function
...
■
■
■
■
Identify common functions
...
● Identify and graph points of discontinuity
...
Understand that even functions have graphs that are
symmetric about the y-axis
...
Recognizing and Classifying Functions
Common Functions
Point-plotting techniques were introduced in Section 2
...
The nine main
functions you will read about in this section will constitute a “library” of functions that you
should commit to memory
...
Several of these functions have been shown
previously in this chapter, but now we will classify them specifically by name and identify
properties that each function exhibits
...
3, we discussed equations and graphs of lines
...
Instead of
the traditional notation of a line, y ϭ mx ϩ b, we use function notation and classify a
function whose graph is a line as a linear function
...
The domain of a linear function ƒ(x) ϭ mx ϩ b is the set of all real numbers R
...
LINEAR F UNCTION: f (x ) ؍mx ؉ b
S LOPE: m
y-I NTERCEPT: b
ƒ(x) ϭ 2x Ϫ 7
mϭ2
b ϭ Ϫ7
ƒ(x) ϭ Ϫx ϩ 3
m ϭ Ϫ1
bϭ3
ƒ(x) ϭ x
mϭ1
bϭ0
ƒ(x) ϭ 5
mϭ0
bϭ5
287
c03a
...
C O N STANT
F U N CTI O N
f(x) = b
b is any real number
...
The y-intercept corresponds
to the point (0, b)
...
The
range, however, is a single value b
...
Points that lie on the graph of a
constant function ƒ(x) ϭ b are
(Ϫ5, b)
(Ϫ1, b)
(0, b)
(2, b)
(4, b)
...
This special case is called the identity function
...
Both the domain
and the range of the identity function are the set of all real numbers R
...
Square Function
Domain: (–∞, ∞) Range: [0, ∞)
y
(–2, 4)
(–1, 1)
S Q UAR E
f(x) = x2
(2, 4)
(1, 1)
F U N CTI O N
x
The graph of the square function is called a parabola and will be discussed in further detail
in Chapters 4 and 8
...
Because squaring a real number always yields a positive number or zero, the range of the
square function is the set of all nonnegative numbers
...
This graph is contained in quadrants
I and II
...
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3
...
CUBE
Cube Function
Domain: (–∞, ∞) Range: (–∞, ∞)
y
F U N CTI O N
f(x) = x3
10
(2, 8)
The domain of the cube function is the set of all real numbers R
...
Note
that the only intercept is the origin and the cube function is symmetric about the origin
...
The next two functions are counterparts of the previous two functions: square root
and cube root
...
S Q UAR E
f (x) = 1x
x
–5
5
(–2, –8)
–10
Square Root Function
Domain: [0, ∞) Range: [0, ∞)
y
5
R O OT F U N CTI O N
(9, 3)
or f (x) = x1/2
(4, 2)
In Section 3
...
The output of the function will be all real
numbers greater than or equal to zero
...
The graph of this function will be contained in quadrant I
...
1, we stated the domain of the cube root function to be (Ϫϱ, ϱ)
...
This graph is contained in quadrants I and III and
passes through the origin
...
In Section 1
...
Now we shift
our focus to the graph of the absolute value function
...
The domain of the absolute value function is the set of all real numbers R, yet the
range is the set of nonnegative real numbers
...
(2, 2)
x
c03a
...
Reciprocal Function
Domain: (–∞, 0) ഫ (0, ∞)
Range: (–∞, 0) ഫ (0, ∞)
y
R E C I P R O CAL
f(x) = 1
x
F U N CTI O N
f(x) =
1
x
x Z 0
(1, 1)
x
(–1, –1)
The only restriction on the domain of the reciprocal function is that x Z 0
...
The graph of the reciprocal
function illustrates that its range is also the set of all real numbers except zero
...
Even and Odd Functions
Of the nine functions discussed above, several have similar properties of symmetry
...
The identity function, cube function, cube root function, and
reciprocal function are all symmetric with respect to the origin
...
Recall from Section 2
...
The box below summarizes the graphic and algebraic characteristics of
even and odd functions
...
If the result is an equivalent equation, the function is
symmetric with respect to the y-axis
...
In any of these equations, if Ϫx is substituted for x, the
result is the same; that is, ƒ(Ϫx) ϭ ƒ(x)
...
All constant
functions are degree zero and are even functions
...
If the result is the negative of the original function, that is, if ƒ(Ϫx) ϭ Ϫƒ(x), then
the function is symmetric with respect to the origin and, hence, classified as an odd
function
...
In
any of these functions, if Ϫx is substituted for x, the result is the negative of the original
function
...
c03a
...
2 Graphs of Functions
291
Be careful, though, because functions that are combinations of even- and odd-degree
polynomials can turn out to be neither even nor odd, as we will see in Example 1
...
Graph y1 = f(x) = x2 - 3
...
a
...
g(x) ؍x5 ؉ x3
Solution (a):
Original function
...
h(x) ؍x2 ؊ x Classroom Example 3
...
1
Determine whether these
functions are even, odd, or neither
...
ƒ(Ϫx) ϭ (Ϫx) Ϫ 3
Simplify
...
Solution (b):
Original function
...
f (x) = 3x3 + x5
b
...
* f (x) = -(3x 3 + x 5)
d
...
* f (x) = (3x 3 + x 5)2
Even; symmetric with respect to the
y-axis
...
Graph y1 = g(x) = x5 + x3
...
odd b
...
odd
d
...
even
Replace x with Ϫx
...
g(؊x) ؊ ؍x5 ؊ x3 (؊ ؍x5 ؉ x3) ؊ ؍g(x)
Because g(؊x) ؊ ؍g(x), we say that g(x) is an odd function
...
Original function
...
h(Ϫx) ϭ (Ϫx)2 Ϫ (Ϫx)
Simplify
...
Graph y1 = h(x) = x 2 - x
...
In parts (a), (b), and (c), we classified these functions as either even, odd, or neither, using
the algebraic test
...
In part (a), we combined two functions: the square function and the
constant function
...
In part (b), we combined two odd functions: the fifth-power function and
the cube function
...
In part (c), we combined two functions: the square function and the
identity function
...
In this part,
combining an even function with an odd function yields a function that is neither even nor odd
and, hence, has no symmetry with respect to the vertical axis or the origin
...
■
■ YO U R T U R N
Classify the functions as even, odd, or neither
...
ƒ(x) ϭ ƒ x ƒ ϩ 4
b
...
even
b
...
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4:55 PM
Page 292
C H A P T E R 3 Functions and Their Graphs
Increasing and Decreasing
Functions
y
(6, 4)
(–1, 1)
(0, 1)
x
(–2, –2)
(2, –2)
Study Tip
• Graphs are read from left to right
...
Look at the figure in the margin to the left
...
If we start
at the left side of the graph and trace the red curve with our pen, we see that the function
values (values in the vertical direction) are decreasing until arriving at the point (Ϫ2, Ϫ2)
...
The values then
remain constant (y ϭ 1) between the points (Ϫ1, 1) and (0, 1)
...
Beyond the point
(2, Ϫ2), the function values increase again until the point (6, 4)
...
When specifying a function as increasing, decreasing, or constant, the intervals are
classified according to the x-coordinate
...
The graph is classified as decreasing when x is less than Ϫ2 and again when x
is between 0 and 2 and again when x is greater than 6
...
In interval notation, this is summarized as
Decreasing
Increasing
Constant
(- ϱ, -2)ഫ(0, 2)ഫ(6, ϱ)
(-2, -1)ഫ(2, 6)
(Ϫ1, 0)
An algebraic test for determining whether a function is increasing, decreasing, or
constant is to compare the value ƒ(x) of the function for particular points in the
intervals
...
A function ƒ is increasing on an open interval I if for any x1 and x2 in I, where
x1 Ͻ x2, then ƒ(x1) Ͻ ƒ(x2)
...
A function ƒ is decreasing on an open interval I if for any x1 and x2 in I, where
x1 Ͻ x2, then ƒ(x1) Ͼ ƒ(x2)
...
A function f is constant on an open interval I if for any x1 and x2 in I, then
ƒ(x1) ϭ ƒ(x2)
...
The range is the set of all y-values (from bottom to top) that the graph of the function
corresponds to
...
An open dot indicates that the graph terminates there and the point is not included in
the graph
...
(An arrow is used in some books to indicate direction
...
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11/24/11
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Page 293
3
...
2
...
10
a
...
(–5, 7)
b
...
(0, 4)
(–2, 4)
Solution (a):
Domain: [Ϫ5, ϱ)
Range: [0, ϱ)
x
–5
(2, 0)
Solution (b):
y
Reading the graph from left to right, we see that
the graph
■
is constant from the point (Ϫ2, 4) to the
point (0, 4)
...
■
Decreasing
(–2, 4)
Constant
–5
Increasing
x
5
(2, 0)
y
The intervals of increasing and decreasing
correspond to the x-coordinates
...
Domain: (- ϱ, 4]
Range: [-3, 2)
b
...
■
(0, 4)
Decreasing
increases from the point (2, 0) on
...
Find the domain and range
...
Find the interval(s) where f is
c, T, or constant
...
■
5
decreasing on the interval (-5, -2)ഫ(0, 2)
...
x
–5
(2, 0)
5
Constant
Increasing
Decreasing Decreasing
Note: The intervals of increasing or decreasing are defined on open intervals
...
For example, the point x ϭ Ϫ5 is included in the
domain of the function but not in the interval where the function is classified as decreasing
...
Let (x1, y1) and (x2, y2) be two points that lie on the graph of a function ƒ
...
This line is called a secant line
...
The slope of the secant line is used to
represent the average rate of change of the function
...
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Page 294
C H A P T E R 3 Functions and Their Graphs
AVE R AG E
R ATE O F C HAN G E
y
Let (x1, ƒ(x1)) and (x2, ƒ(x2)) be two distinct
points, (x1 Z x2) , on the graph of the function
ƒ
...
2
...
x = 2 to x = -1
3
b
...
* x = a to x = 2a, where a
is a positive real number
Answer:
a
...
-
14
8
3
c
...
x ϭ Ϫ1 to x ϭ 0
b
...
x ϭ 1 to x ϭ 2
Solution (a):
Write the average rate of change formula
...
=
f(0) - f(-1)
0 - (-1)
Substitute ƒ(Ϫ1) ϭ (Ϫ1)4 ϭ 1 and
ƒ(0) ϭ 04 ϭ 0
...
ϭ -1
Solution (b):
Write the average rate of change formula
...
=
f(1) - f(0)
1 - 0
Substitute ƒ(0) ϭ 04 ϭ 0 and
ƒ(1) ϭ (1)4 ϭ 1
...
ϭ 1
Solution (c):
Write the average rate of change formula
...
=
f(2) - f (1)
2 - 1
Substitute ƒ(1) ϭ 14 ϭ 1 and
ƒ(2) ϭ (2)4 ϭ 16
...
ϭ 15
c03b
...
2 Graphs of Functions
Graphical Interpretation: Slope of the Secant Line
y
a
...
Average rate of change of f from x ϭ 0 to x ϭ 1:
Increasing at a rate of 1
3
(1, 1)
x
(0, 0)
–2
2
–2
y
c
...
x ϭ Ϫ2 to x ϭ 0
b
...
W OR DS
Let the difference between
x1 and x2 be h
...
Substitute x2 Ϫ x1 ϭ h into
the denominator and
M ATH
x2 Ϫ x1 ϭ h
x2 ϭ x1 ϩ h
Average rate of change =
f(x2) - f(x1)
x2 - x1
x2 ϭ x1 ϩ h into the numerator
of the average rate of change
...
=
f(x + h) - f(x)
h
■
Answer: a
...
2
295
c03b
...
f (x + h)
DEFINITION
The expression
f (x)
Difference Quotient
f(x + h) - f(x)
, where h Z 0, is called the difference quotient
...
In calculus the difference
quotient is used to define a derivative
...
f (x + h) - [ f (x)]
h
Solution:
Find ƒ(x ϩ h)
...
2
...
y
f(x + h)
f(x)
4
Find the difference
quotient
...
f (x + h) - f (x)
2x2 + 4xh + 2h2 + 1 - 2x2 - 1
=
h
h
2
Answer: -6x - 3h
f (x + h) - f (x)
4xh + 2h2
=
h
h
Factor the numerator
...
■
Answer:
f (x + h) - f (x)
h
= - 2x - h
f (x + h) - f (x)
h(4x + 2h)
=
h
h
f (x + h) - f (x)
= 4x + 2h
h
■ YO U R T U R N
hZ0
Calculate the difference quotient for the function ƒ(x) ϭ Ϫx2 ϩ 2
...
Sometimes the need arises to define functions in terms of pieces
...
For instance, if
a particular plumber charges $100 to drive out to your house and work for 1 hour and then an
additional $25 an hour for every additional hour he or she works on your job, we would define
this function in pieces
...
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3
...
Another piecewise-defined function is the absolute value function
...
We start by graphing these two lines on the same graph
...
Absolute value function
f(x) = ƒ xƒ = b
-x
x
297
f (x)
y = –x
y=x
x
x<0
x » 0
f (x)
f (x) = | x |
The next example is a piecewise-defined function given in terms of functions in our
“library of functions
...
This is like the procedure above for the absolute
value function
...
Solution:
x2
G(x) = c 1
x
x 6 -1
-1 … x … 1
x 7 1
Constant function:
f (x) 1 ؍
Plot a piecewise-defined function
using the TEST menu operations
to define the inequalities in the
function
...
Square function:
f(x) ؍x2
x
(Ϫ) 1 ) ϩ ( 1 )
( X, T, , n
2nd MATH Test 4 Ն (Ϫ) 1
2
f (x) = 1
1
(–1, 1)
–2
6 Յ 1 ) ϩ ( X, T, , n ) (
(1, 1)
–1
1
) ( X, T, , n 2nd MATH Test
x
X, T, , n 2nd MATH Test 3
2
Ͼ 1 )
Identity function:
f(x) ؍x
The points to focus on in particular are the x-values
where the pieces change over—that is, x ϭ Ϫ1 and
x ϭ 1
...
When x Ͻ Ϫ1, this
function is defined by the square function, ƒ(x) ؍x2, so
darken that particular function to the left of x ϭ Ϫ1
...
When x Ͼ 1,
the function is defined by the identity function, ƒ(x) ؍x,
so darken that function to the right of x ϭ 1
...
y
5
Set the viewing rectangle as [Ϫ4, 4]
by [Ϫ2, 5]; then press GRAPH
...
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Technology Tip
This function is defined for all real values of x, so the domain of this function is the set of
all real numbers
...
Hence, the range of this function is [1, ϱ)
...
Press:
Decreasing: (- ϱ, -1)
Constant: (- 1, 1)
Increasing: (1, ϱ)
Yϭ ( 1 Ϫ X, T, , n ) (
X, T, , n 2nd MATH 5 Ͻ 0
) ϩ (
X, T, , n ) ( X, T, , n
2nd MATH 4 Ն 0 ) (
X, T, , n 2nd MATH 5 Ͻ 2
) ϩ (
(Ϫ) 1 )
( X, T, , n
2nd MATH 3 Ͼ 2 )
The term continuous implies that there are no holes or jumps and that the graph can be
drawn without picking up your pencil
...
The previous example illustrates a continuous piecewise-defined function
...
At
the x ϭ 1 junction, the constant function and the identity function both pass through the
point (1, 1)
...
The next example illustrates a discontinuous piecewise-defined function
...
To avoid connecting graphs of the
pieces, press MODE and Dot
...
Solution:
1 - x
f (x) = c x
-1
x 6 0
0 … x 6 2
x 7 2
y
Graph these functions on the same plane
...
The table of values supports the graph,
except at x ϭ 2
...
Constant function:
ƒ(x) 1؊ ؍
Darken the piecewise-defined function on the graph
...
Note the use of an open circle, indicating up to
but not including x ϭ 0
...
The circle is filled in at the left endpoint, x ϭ 0
...
For all
values greater than 2, x>2, the function is defined by the constant function
...
y
x
c03b
...
2 Graphs of Functions
299
At what intervals is the function increasing, decreasing, or constant? Remember that the
intervals correspond to the x-values
...
Domain: (- ϱ, 2)ഫ(2, ϱ)
The output of this function (vertical direction) takes on the y-values y Ն 0 and the additional
single value y ϭ Ϫ1
...
In
this example, the point x ϭ 0 corresponds to a jump, because you would have to pick up your
pencil to continue drawing the graph
...
The hole indicates that the function is not defined at that point, and there is still a jump because
the identity function and the constant function do not meet at the same y-value at x ϭ 2
...
-x
f(x) = c 2
x
■
Answer: Increasing: (1, ϱ)
Decreasing: (Ϫϱ, Ϫ1)
Constant: (Ϫ1, 1)
Domain:
(- ϱ, 1) ഫ (1, ϱ)
Range: [1, ϱ)
y
x … -1
-1 6 x 6 1
x 7 1
x
Piecewise-defined functions whose “pieces” are constants are called step functions
...
A
common step function used in engineering is the Heaviside step function (also called the
unit step function):
H(t) = b
0
1
t 6 0
t Ú 0
1
This function is used in signal processing to represent a signal that turns on at some time
and stays on indefinitely
...
G R E ATE ST
I NTE G E R F U N C T I O N
f(x) = [[x]] = greatest integer less than or equal to x
...
0
1
...
5
1
...
9
2
...
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SECTION
3
...
■
Inspect the graph to determine the set of all inputs (domain)
and the set of all outputs (range)
...
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3
...
Decreasing: Graph of function falls from left to right
...
Average Rate of Change
h Z 0
Piecewise-Defined Functions
■
■
f (x2) - f (x1)
x2 - x1
f (x + h) - f (x)
h
Difference Quotient
301
Continuous: You can draw the graph of a function without
picking up the pencil
...
x1 Z x2
SECTION
3
...
1
...
h(x) ϭ 3 Ϫ x
3
...
F(x) ϭ x4 ϩ 2x2
5
...
ƒ(x) ϭ 3x5 ϩ 4x3
7
...
G(x) ϭ 2x4 ϩ 3x3
18
...
G(t) = 1t - 3
1
+ 3
x
23
...
20
...
ƒ(x) ϭ ƒ x ƒ ϩ 5
14
...
ƒ(x) ϭ ƒ x ƒ
21
...
g(x) = 2x + x
16
...
g(x) ϭ xϪ1 ϩ x
9
...
G(t) ϭ ƒ t Ϫ 3 ƒ
22
...
ƒ(x) ϭ ƒ x ƒ ϩ x2
2
1
- 2x
x
y
y
10
9
x
x
–5
–5
5
5
In Exercises 25–36, state the (a) domain, (b) range, and (c) x-interval(s) where the function is increasing, decreasing, or constant
...
25
...
27
...
y
y
5
f (x)
f (x)
(–3, 4) 5
(0, 4)
(–3, 3)
(–3, 3)
x
(–2, 2)
(2, 1)
(–7, 1)
x
x
2
–8
(–2, –1)
y
y
(–1, –1)
(0, –1)
(–6, 0)
–10
(6, 0) x
10
(0, 0)
(1, –1)
(–4, –2)
–5
–5
(2, –5)
–5
(3, –4)
c03b
...
30
...
y
32
...
36
...
y
x
–5 (–3, 0)
–5 (0, –4)
–5
33
...
ƒ(x) ϭ x2 Ϫ x
f(x ؉ h) ؊ f(x)
for each function
...
ƒ(x) ϭ x2 ϩ 2x
39
...
ƒ(x) ϭ 5x Ϫ x2
41
...
ƒ(x) ϭ x2 Ϫ 2x ϩ 5
44
...
ƒ(x) ϭ Ϫ3x2 ϩ 5x Ϫ 4
In Exercises 45–52, find the average rate of change of the function from x 1 ؍to x
...
ƒ(x) ϭ x3
46
...
ƒ(x) ϭ 1 Ϫ 2x
47
...
ƒ(x) ϭ 9 Ϫ x2
51
...
f (x) = 2x2 - 1
48
...
State the domain and range in interval notation
...
53
...
f (x) = b
59
...
f(x) = d
3
4 + x
2
3 -
x 6 1
x Ú1
x 6 -2
x 7 -2
54
...
f (x) = b
60
...
G(x) = c x
3
x … -1
x 7 -1
x 6 -1
-1 … x … 3
x 7 3
55
...
f (x) = b
61
...
G(x) = c x
3
x 6 2
x 7 2
x 6 -1
-1 6 x 6 3
x 7 3
c03b
...
2 Graphs of Functions
1
65
...
f (x) = c x + 1
-x + 1
69
...
G(x) = c 1
x
1
3
1x
72
...
f (x) = c 1
ƒ xƒ
x … -2
-2 6 x 6 2
xÚ2
x2
78
...
G(x) = c x
1x
x Z 0
x + 3
75
...
G(x) = b
x = 0
3
t 6 1
1 6 t 6 2
t 7 2
-x - 1
68
...
G(x) = c x
- 1x
x
77
...
G(t) = c t2
4
t 6 1
1 … t … 2
t 7 2
303
3
x 6 -1
-1 … x 6 1
x 7 1
x 6 -1
-1 6 x 6 1
x 7 1
x … -1
-1 6 x 6 1
xÚ1
A P P L I C AT I O N S
For Exercises 79 and 80, refer to the following:
A manufacturer determines that his profit and cost functions over
one year are represented by the following graphs
...
Business
...
80
...
Find the intervals on which cost is increasing,
decreasing, and constant
...
Budget: Costs
...
If the sorority orders 50 or
fewer T-shirts, the cost is $10 per shirt
...
If it
orders more than 100, the cost is $8 per shirt
...
82
...
The marching band at a university is
ordering some additional uniforms to replace existing
uniforms that are worn out
...
12 per uniform
...
73
per uniform
...
83
...
The Richmond rowing club is planning to
enter the Head of the Charles race in Boston and is trying to
figure out how much money to raise
...
Find the cost function C(x) as a function of the number
of boats x the club enters
...
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84
...
A phone company
charges $
...
12 per minute
every minute after that
...
85
...
A young couple are planning their wedding
reception at a yacht club
...
The cost
of food is $35 per person for the first 100 people and $25 per
person for every additional person beyond the first 100
...
91
...
The following table corresponds to
first-class postage rates for the U
...
Postal Service
...
WEIGHT LESS
THAN (OUNCES)
F IRST-C LASS R ATE
(F LAT E NVELOPES)
1
$0
...
97
3
$1
...
Home Improvement
...
The parts are
$1400, and the labor is $25 per hour
...
4
$1
...
48
6
$1
...
82
87
...
A famous author negotiates with her publisher the
monies she will receive for her next suspense novel
...
If the book sells for $20 and royalties are based on the selling
price, write a royalties function R(x) as a function of total
number x of books sold
...
99
9
$2
...
33
11
$2
...
67
13
$2
...
Sales
...
89
...
Some artists are trying to decide whether they
will make a profit if they set up a Web-based business to
market and sell stained glass that they make
...
The materials cost $35 for each work in stained glass,
and the artists charge $100 for each unit they sell
...
92
...
The following table corresponds to
first-class postage rates for the U
...
Postal Service
...
F IRST-C LASS R ATE
(PARCELS)
1
$1
...
Profit
...
He
orders gulf shrimp to be flown in from New Orleans
...
The shipping costs $30
...
Assume that each person will eat 1 pound of
shrimp
...
30
3
$1
...
64
5
$1
...
98
7
$2
...
32
9
$2
...
66
11
$2
...
00
13
$3
...
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3
...
A square wave alternates regularly and
instantaneously between two levels
...
Climate Change: Global Warming
...
1900 to 1950?
b
...
Electronics: Signals
...
96
...
What is the average
rate of change in global carbon emissions from fossil fuel
burning from
a
...
1975 to 2000?
f (t)
5
For Exercises 97 and 98, use the following information:
t
5
The height (in feet) of a falling object with an initial velocity of 48
feet per second launched straight upward from the ground is given
by h(t) ϭ Ϫ16t2 ϩ 48t, where t is time (in seconds)
...
Falling Objects
...
Falling Objects
...
Electronics: Signals
...
For Exercises 99 and 100, refer to the following:
An analysis of sales indicates that demand for a product during a
calendar year (no leap year) is modeled by
1
d(t) = 32t2 + 1 - 2
...
99
...
Find the average rate of change of the demand
of the product over the first quarter
...
Economics
...
c03b
...
101
...
State the domain and
range
...
What
mistake was made?
103
...
Write a function describing the cost of the service as a
function of minutes used online
...
This gives
us the familiar absolute
value graph
...
What mistake was made?
Domain: (Ϫϱ, ϱ) or R
Range: [0, ϱ)
This is incorrect
...
Graph the piecewise-defined function
...
Solution:
y
The resulting graph is
as shown
...
f(x) = b
Darken the function
ƒ(x) ؊ ؍x when x Ͻ 1
and the function ƒ(x) ؍x
when x Ͼ 1
...
Most money market accounts pay a higher interest with a
higher principal
...
Solution: I(x) = b
0
...
02(10,000) + 0
...
What mistake was made?
Draw the graphs of
ƒ(x) ؊ ؍x and
ƒ(x) ؍x
...
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3
...
105
...
106
...
107
...
108
...
■
CHALLENGE
In Exercises 109 and 110, for a and b real numbers, can the function given ever be a continuous function? If so, specify the value
for a and b that would make it so
...
f (x) = b 2
110
...
In trigonometry you will
learn about the sine
function, sin x
...
It should
look like the graph on the
–10
right
...
In trigonometry you will learn about the tangent function,
tan x
...
If
p
p
you restrict the values of x so that - 6 x 6 , the graph
2
2
should resemble the graph below
...
In trigonometry you will
learn about the cosine
function, cos x
...
It
should look like the graph
on the right
...
5
1
x
–10
1
...
Plot the function f(x) =
–1
sin x
...
Graph the function f (x) = [[3x]] using a graphing utility
...
116
...
3
State the domain and range
...
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SECTION
3
...
Sketch the graph of a function by reflecting a common
function about the x-axis or y-axis
...
Sketch the graph of a function using a sequence of
transformations
...
Apply multiple transformations of common functions
to obtain graphs of functions
...
Horizontal and Vertical Shifts
The focus of the previous section was to learn the graphs that correspond to particular
functions such as identity, square, cube, square root, cube root, absolute value, and reciprocal
...
In this section, we will
x
discuss how to sketch the graphs of functions that are very simple modifications
of these functions
...
Collectively, these techniques are called
transformations
...
The graph of ƒ(x) ͦ ؍x ͦ was given in
the last section
...
Graphing these functions by point-plotting yields
y
x
2
؊1
h(x)
f (x)
ƒ(x)
؊2
g (x)
1
x
g(x)
x
h(x)
؊2
4
؊2
3
؊1
3
؊1
2
1
0
0
0
2
1
1
1
3
1
0
2
x
0
2
2
4
2
1
Instead of point-plotting the function g(x) ͦ ؍x ͦ ؉ 2, we could have started with
the function f(x) ͦ ؍x ͦ and shifted the entire graph up 2 units
...
In both cases, the base or starting function is f(x) ͦ ؍x ͦ
...
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3
...
Therefore, the output for g(x) is two more than the typical
output for ƒ(x)
...
In general, shifts that occur outside the function correspond to a vertical
shift corresponding to the sign of the shift
...
In the case of h(x), the shift occurs “inside” the function—that is, inside the parentheses
showing the argument
...
The y-value remained the same, but the x-value
shifted to the right one unit
...
In general, shifts that occur inside the function correspond
to a horizontal shift opposite the sign
...
If, instead, we had the function
H(x) ϭ ƒ x ϩ 1 ƒ , this graph would have started with the graph of the function ƒ(x) and shifted to
the left one unit
...
Up (ϩ)
Down (Ϫ)
Adding or subtracting a constant outside the function corresponds to a vertical shift
that goes with the sign
...
Shifts inside the function are
horizontal shifts opposite the sign
...
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C H A P T E R 3 Functions and Their Graphs
EXAMPLE 1
Horizontal and Vertical Shifts
Technology Tip
a
...
a
...
H(x) ϭ (x ϩ 1)2
(1, 1)
(0, 0)
–2
x
2
Solution:
In both cases, the function to start with is ƒ(x) ؍x2
...
Graphs of y1 = x2 and
y2 = H(x) = (x + 1)2 are shown
...
b
...
1
...
Therefore, we expect a horizontal shift that goes
opposite the sign
...
Since the sign is positive, this corresponds to a
shift to the left
...
Shifting the graph of the function ƒ(x) ؍x2 to
the left one unit yields the graph of
H(x) ( ؍x ؉ 1)2
...
y
a
...
1
...
Therefore, we expect a vertical shift
that goes with the sign
...
Since the sign is negative, this corresponds to
a downward shift
...
Shifting the graph of the function ƒ(x) ؍x2
down one unit yields the graph of
g(x) ؍x2 ؊ 1
...
a
...
H(x) ϭ (x Ϫ 1)2
y
10
x
–5
5
It is important to note that the domain and range of the resulting function can be thought of
as also being shifted
...
c03b
...
3 Graphing Techniques: Transformations
EXAMPLE 2
a
...
G(x) = 1x - 2
311
Classroom Example 3
...
2
Graph these using translation and
state the domain and range
...
f (x) = ƒ x ƒ - 2
b
...
y
5
Answer: The graphs are given by:
In both cases the function to start with is f (x) 1 ؍x
...
g(x) 1 ؍x ؉ 1 can be rewritten as
g(x) ϭ ƒ(x ϩ 1)
...
The shift (one unit) is inside the function,
which corresponds to a horizontal
shift opposite the sign
...
Shifting the graph of f (x) 1 ؍x
to the left one unit yields the graph
of g(x) 1 ؍x ؉ 1
...
(0, 0)
10
y
5
a
...
Domain: [0, ϱ), Range: [2, ϱ)
(8, 3)
(3, 2)
(0, 1)
Although the original function f (x) 1 ؍x had an implicit restriction on the domain:
[0, ؕ), the function g(x) 1 ؍x ؉ 1 has the implicit restriction that x Ն Ϫ1
...
b
...
Domain:
[؊1, ؕ)
Range:
x
9
(Ϫ1, 0)
[0, ؕ)
■
y
(9, 3)
3
2
...
Note
that the point (0, 0), which lies on the graph
of ƒ(x), gets shifted to the point (0, ؊2) on
the graph of G(x)
...
The shift (two units) is outside the function,
which corresponds to a vertical shift with
the sign
...
G(x) = 1x - 2
–2
8
10
The original function f (x) 1 ؍x has an implicit restriction on the domain: [0, ؕ)
...
The output or range of
G(x) is always two units less than the output of the original function ƒ(x)
...
G(x) = 1x - 2
–2 (0, Ϫ2)
–5
Domain: [2, ϱ)
Range: [0, ϱ)
b
...
b
...
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C H A P T E R 3 Functions and Their Graphs
The previous examples have involved graphing functions by shifting a known function
either in the horizontal or vertical direction
...
EXAMPLE 3
Combining Horizontal and Vertical Shifts
Sketch the graph of the function F(x) ϭ (x ϩ 1)2 Ϫ 2
...
Technology Tip
a
...
Solution:
The base function is y ؍x2
...
The shift (one unit) is inside the function, so it represents a horizontal shift opposite
the sign
...
The Ϫ2 shift is outside the function, which
represents a vertical shift with the sign
...
Therefore, we shift the graph of y ؍x2
8
to the left one unit and down two units
...
Domain: (؊ؕ, ؕ)
Range: [؊2, ؕ)
x
–5
5
(–1, –2)
■
Answer:
ƒ(x) ͦ ؍x ؊ 2 ͦ ؉1
ƒ(x) ͦ ؍x ͦ
Domain: (Ϫϱ, ϱ)
Range: [1, ϱ)
■ YO U R T U R N
Sketch the graph of the function ƒ(x) ϭ ƒ x Ϫ 2 ƒ ϩ 1
...
y
All of the previous transformation examples involve starting with a common function
and shifting the function in either the horizontal or the vertical direction (or a combination
of both)
...
10
Reflection about the Axes
x
–5
5
To sketch the graphs of ƒ(x) ؍x2 and g(x) ؊ ؍x2 start by first listing points that are on each
of the graphs and then connecting the points with smooth curves
...
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3
...
Also note that the function g(x) can be written as the negative of the
function ƒ(x); that is, g(x) ϭ Ϫƒ(x)
...
Let’s now investigate reflection about the y-axis
...
x
f(x)
x
0
0
؊9
1
؊4
2
؊1
1
9
3
0
Answer: The graphs are given by:
y
2
4
Classroom Example 3
...
3
Graph using translation and state
the domain and range
...
f (x) = 1x - 1 - 1
b
...
h(x) = (x - 1)3 + 2
3
1
313
g(x)
0
5
g
Note that if the graph of f (x) 1 ؍x is reflected about the y-axis, the result is the graph of b
...
Also note that the function g(x) can be written as g(x) ϭ ƒ(Ϫx)
...
Domain: (- ϱ, ϱ), Range: (- ϱ, ϱ)
reflection about the y-axis is produced by replacing x with –x in the function
...
x
–5
5
a
...
The graph of ƒ(؊x) is obtained by reflecting the graph of ƒ(x) about the y-axis
...
EXAMPLE 4
Sketching the Graph of a Function Using Both
Shifts and Reflections
Solution:
Start with the square root function
...
Reflect the graph of ƒ(x ϩ 1) about the
x-axis to arrive at the graph of Ϫƒ(x ϩ 1)
...
3
...
f (x) = - ƒ x - 2ƒ
b
...
h(x) = - (x + 1)2
f (x) 1 ؍x
f(x ؉ 1) 1 ؍x ؉ 1
Answer: The graphs are given
by:
؊f(x ؉ 1) 1 ؊ ؍x ؉ 1
y
5
x
9
–5
c03b
...
Graphs of y1 = 1x, y2 = 1x + 2,
y3 = 1-x + 2, and
y4 = f(x) = 12 - x + 1 are
shown
...
EXAMPLE 5
Sketching the Graph of a Function Using Both
Shifts and Reflections
Solution:
Start with the square root function
...
Reflect the graph of g(x ϩ 2) about the
y-axis to arrive at the graph of g(Ϫx ϩ 2)
...
g(x) 1 ؍x
g(x ؉ 2) 1 ؍x ؉ 2
g(؊x ؉ 2) ؊1 ؍x ؉ 2
g(؊x ؉ 2) ؉ 1 ؊ 21 ؍x ؉ 1
y
5
x
■
Answer:
Domain: [1, ϱ)
Range: (Ϫϱ, 2]
Use shifts and reflections to sketch the graph of the function
f(x) = - 2x - 1 + 2
...
–5
y
■ YO U R T U R N
5
x
10
Start with the square root function
...
Classroom Example 3
...
5*
Graph:
a
...
g(x) = - ( x - 1)3 + 2
c
...
Let us consider an alternate order of transformations
...
Replace x with x Ϫ 2, which corresponds
to a shift of the graph of g(Ϫx) ϩ 1 to
the right two units to arrive at the graph of
g[Ϫ(x Ϫ 2)] ϩ 1
...
To avoid any possible confusion, follow this
order of transformations:
1
...
Reflection: ƒ(Ϫx) and/or Ϫƒ(x)
3
...
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Page 315
3
...
These transformations
(shifts and reflections) are called rigid transformations because they alter only the
position
...
We now consider stretching and compressing of graphs in both the vertical and the
horizontal direction
...
For example, the graphs of the functions ƒ(x) ؍x2, g(x) 2 ؍ƒ(x) 2 ؍x2,
and h(x) 1 ؍f(x) 1 ؍x2 are illustrated below
...
y
20
x
ƒ(x)
x
g(x)
x
h(x)
؊2
4
؊2
8
؊2
2
؊1
1
؊1
2
؊1
1
2
0
0
0
0
0
0
1
1
1
2
1
1
2
2
4
2
8
2
2
x
–5
5
Note that when the function ƒ(x) ؍x2 is multiplied by 2, so that g(x) 2 ؍ƒ(x) 2 ؍x2, the
result is a graph stretched in the vertical direction
...
V E RTI CAL
STR E TC H I N G AN D VE RTI CAL
C O M P R E S S I N G O F G R AP H S
The graph of cf(x) is found by:
■ Vertically stretching the graph of ƒ(x)
■ Vertically compressing the graph of ƒ(x)
Note: c is any positive real number
...
4
Solution:
1
...
ƒ(x) ؍x3
2
...
4
h(x) ؍
1 3
x
4
315
c03b
...
3
...
f (x) =
1
2 ƒ xƒ
1
2ƒx -
1ƒ
b
...
h(x) = 2 ƒ x ƒ
d
...
Determine a few points that lie on
the graph of h
...
If c Ͼ 1, then the result is a
horizontal compression of the graph of ƒ
...
EXAMPLE 7
if 0 Ͻ c Ͻ 1
if c Ͼ 1
Vertically Stretching and Horizontally
Compressing Graphs
Given the graph of ƒ(x), graph:
a
...
2ƒ(x)
y
b
...
g = 2 1 x
2
Classroom Example 3
...
7
Graph:
2
1
c
...
2
( , 2)
2
1
x
5
2
–1
–2
( 3 , –2)
2
c03b
...
3 Graphing Techniques: Transformations
Solution (b):
317
y
Since the argument of the function is
multiplied (on the inside) by 2, the result
is that each x-value of ƒ(x) is divided by 2,
which corresponds to horizontal
compression
...
y
40
g(x)
x
f (x)
–2
■ YO U R T U R N
EXAMPLE 8
2
–40
Graph the function g(x) ϭ 4x3
...
Technology Tip
Graphs of y1 = x2, y2 = (x - 3)2,
y3 = 2(x - 3)2, and
y4 = H(x) = -2(x - 3)2 are shown
...
ƒ(x) ؍x2
Shift the graph of f (x) to the right three
units to arrive at the graph of ƒ(x Ϫ 3)
...
2ƒ(x ؊ 3) ϭ 2(x ؊ 3)2
Reflect the graph 2 f (x Ϫ 3) about the
x-axis to arrive at the graph of Ϫ2ƒ(x Ϫ 3)
...
g = 1 1-x - 1
2
Classroom Example 3
...
8*
Graph:
6
–5
In Example 8 we followed the same “inside out” approach with the functions to determine
the order for the transformations: horizontal shift, vertical stretch, and reflection
...
h = - 1 21 x + 1
2
2
a
...
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C H A P T E R 3 Functions and Their Graphs
SECTION
3
...
Shift the graph of ƒ to the right c units
...
Replace x by x Ϫ c
...
Shift the graph of ƒ down c units
...
Subtract c from ƒ(x)
...
Multiply ƒ(x) by Ϫ1
...
Replace x by Ϫx
...
Multiply ƒ(x) by c
...
Multiply ƒ(x) by c
...
Replace x by cx
...
Replace x by cx
...
3
■
EXERCISES
SKILLS
7
...
8
...
f (x) = 1-x + 1
11
...
f (x) = - 11 - x - 1
b
...
f (x) = 11 - x - 1
1
...
d
...
ƒ(x) ϭ (x Ϫ 1)2
6
...
ƒ(x) ϭ Ϫ(x ϩ 1)2
a
...
ƒ(x) ϭ Ϫx2 Ϫ 1
3
...
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Page 319
3
...
f
...
h
...
k
...
i
...
13
...
Shifted to the left four units
15
...
Reflected about the x-axis
17
...
Vertically compressed by a factor of 3
In Exercises 19–24, write the function whose graph is the graph of y ؍x3, but is transformed accordingly
...
Shifted down four units
20
...
Shifted up three units and to the left one unit
22
...
Reflected about the y-axis
24
...
25
...
y ϭ f (x Ϫ 2)
b
...
y ϭ f (x ϩ 2)
b
...
27
...
a
...
y ϭ f (x Ϫ 3)
a
...
y ϭ f (x ϩ 3)
c03b
...
31
...
y
y
x
a
...
y ϭ f (Ϫx)
32
...
y ϭ Ϫf (x)
b
...
y ϭ f(x Ϫ 2) Ϫ 3
x
a
...
y ϭ f (2x)
x
a
...
y ϭ f (2x)
37
...
y ϭ f(x ϩ 1) Ϫ 2
y
f (x)
y
g(x)
38
...
y ϭ Ϫf(x Ϫ 1) ϩ 2
39
...
y ϭ Ϫ2f(x) ϩ 1
x
y
41
...
y = 2G(x + 1) - 4
F(x)
42
...
y = gA 1x B
2
y
46
...
y = -2G(x - 1) + 3
43
...
y = -G(x - 2) - 1
x
44
...
49
...
y ϭ x2 ϩ 3
51
...
y ϭ (x Ϫ 2)2
53
...
y ϭ (x ϩ 2)2 ϩ 1
55
...
y ϭ Ϫ(x ϩ 2)2
57
...
y ϭ Ϫ ƒ x ƒ
59
...
y ϭ ƒ 1 Ϫ x ƒ ϩ 2
61
...
y ϭ 2 ƒ x ƒ ϩ 1
65
...
y =
1
+ 2
x + 3
73
...
y = 12 - x + 3
74
...
y =
1
3 - x
63
...
y = 1x - 1 + 2
71
...
y = 12 - x
3
68
...
y = 2 -
1
1 - x
c03b
...
3 Graphing Techniques: Transformations
321
In Exercises 75–80, transform the function into the form f(x) ؍c(x ؊ h)2 ؉ k, where c, k, and h are constants, by completing the
square
...
75
...
f (x) ϭ x2 ϩ 2x Ϫ 2
79
...
f(x) ϭ 3x2 Ϫ 6x ϩ 5
■
77
...
f(x) ϭ Ϫx2 ϩ 6x Ϫ 7
A P P L I C AT I O N S
81
...
A manager hires an employee at a rate of $10 per
hour
...
After a year, the manager decides to award the
employee a raise equivalent to paying him for an additional
5 hours per week
...
82
...
The profit associated with St
...
In rainy years Sod King gives away 10 free pallets per
year
...
83
...
Every year in the United States each working
American typically pays in taxes a percentage of his or her
earnings (minus the standard deduction)
...
22(x Ϫ 6500)
...
Write the function that will
determine her 2012 taxes, assuming she receives the raise
that places her in the 33% bracket
...
BSA can be modeled by the function
BSA =
wh
A 3600
where w is weight in kilograms and h is height in centimeters
...
However, for an individual
adult height is generally considered constant; thus BSA can be
thought of as a function of weight alone
...
Health/Medicine
...
(b) If she loses
3 kilograms, find a function that represents her new BSA
...
Health/Medicine
...
(b) If he gains
5 kilograms, find a function that represents his new BSA
...
Medication
...
The
number of milliliters of an antiseizure medication A is
given by A(x) = 1x + 2, where x is the weight of the
infant in ounces
...
What is the function that represents the
actual amount of medication the infant is given if his
weight is overestimated by 3 ounces?
■
C AT C H T H E M I S TA K E
87
...
In Exercises 87–90, explain the mistake that is made
...
Start with the function f (x) = 1x
...
Shift the function to the left three units
...
Shift the function up two units
...
What mistake was made?
88
...
Solution:
a
...
b
...
c
...
d
...
This is incorrect
...
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C H A P T E R 3 Functions and Their Graphs
89
...
90
...
Solution:
a
...
b
...
c
...
d
...
This is incorrect
...
Start with the function f(x) ϭ x2
...
Reflect the function about the y-axis
...
Shift the function up one unit
...
Expand in the vertical direction by a factor of 2
...
What mistake was made?
CONCEPTUAL
92
...
In Exercises 91–94, determine whether each statement is true or false
...
The graph of y ϭ ƒ Ϫx ƒ is the same as the graph of y ϭ ƒ x ƒ
...
The point (a, b) lies on the graph of the function y ϭ f(x)
...
If the graph of an odd function is reflected about the x-axis
and then the y-axis, the result is the graph of the original odd
function
...
If the graph of y = 1 is reflected about the x-axis, it produces
x
the same graph as if it had been reflected about the y-axis
...
Use a graphing utility to graph:
a
...
y ϭ x Ϫ 1 and y ϭ ƒ x ϩ 1 ƒ
3
3
What is the relationship between ƒ(x) and ƒ ƒ(x) ƒ ?
98
...
y ϭ x2 Ϫ 2 and y ϭ ƒ x ƒ 2 Ϫ 2
b
...
y = 1x and y = 10
...
y = 1x and y = 110x
99
...
The point (a, b) lies on the graph of the function y ϭ ƒ(x)
...
y = 1x and y = 0
...
y = 1x and y = 10 1x
100
...
Use a graphing utility to graph y = f (x) = [[0
...
Use transforms to describe the relationship between f(x) and
y = [[ x]]
...
Use a graphing utility to graph y = g(x) = 0
...
Use transforms to describe the relationship between g(x) and
y = [[ x]]
...
qxd
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SECTION
3
...
Evaluate composite functions
...
■
■
Understand domain restrictions when dividing
functions
...
Two different functions can be combined using mathematical operations such as addition,
subtraction, multiplication, and division
...
When we combine
functions, we do so algebraically
...
Adding, Subtracting, Multiplying,
and Dividing Functions
Consider the two functions f(x) ϭ x2 ϩ 2x Ϫ 3 and g(x) ϭ x ϩ 1
...
Therefore, we can add, subtract, or multiply
these functions for any real number x
...
Subtraction: ƒ(x) Ϫ g(x) ϭ x2 ؉ 2x ؊ 3 Ϫ (x ؉ 1) ϭ x2 ϩ x Ϫ 4
The result is in fact a new function, which we denote:
( f Ϫ g) (x) ϭ x2 ϩ x Ϫ 4
This is the difference function
...
Although both f and g are defined for all real numbers x, we must restrict x so that x Z -1 to
f
form the quotient
...
Two functions can be added, subtracted, and multiplied
...
However, for division, any value
of x (input) that makes the denominator equal to zero must be eliminated from the domain
...
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C H A P T E R 3 Functions and Their Graphs
The previous examples involved polynomials
...
Adding, subtracting, and multiplying polynomials result in other
polynomials, which have domains of all real numbers
...
The domain of the sum function, difference function, or product function is the
intersection of the individual domains of the two functions
...
However, any values that
make the denominator zero must also be eliminated
...
The exception to this is the quotient
function, which also eliminates values that make the denominator equal to zero
...
4
...
Compute
f
f + g, f - g, fg, and g
...
Answer:
3
f + g = 1 - 2 1x
Domain: ( - ϱ, ϱ),
For the functions f (x) = 2x - 1 and g(x) = 24 - x, determine the sum function,
difference function, product function, and quotient function
...
EXAMPLE 1
Solution:
Sum function:
Difference function:
3
f - g = 1 + 61x
Domain: ( - ϱ, ϱ),
3
Product function:
3
fg = - 41x - 82x2
Domain: ( - ϱ, ϱ),
3
f
1 + 21x
=
3
g
- 41x
Domain: x Z 0
...
Domain of ƒ(x): [1, ϱ)
Domain of g(x): (- ϱ, 4]
The domain of the sum, difference, and product functions is
■
Answer:
( f + g)(x) = 2x + 3 + 21 - x
Domain: [- 3, 1]
[1, ϱ) പ (- ϱ, 4] = [1, 4]
Given the function f (x) = 1x + 3 and g(x) = 11 - x, find
( f ϩ g)(x) and state its domain
...
This
implies that x Z 4, so the domain of the quotient function is [1, 4)
...
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Page 325
Given the functions F(x) = 1x and G(x) ϭ ƒ x Ϫ 3 ƒ , find the quotient function,
F
Classroom Example 3
...
2
a b(x), and state its domain
...
Form g and
The quotient function is written as
state the domain
...
4 Operations on Functions and Composition of Functions
EXAMPLE 2
Quotient Function and Domain Restrictions
a
F(x)
1x
F
=
b(x) =
G
G(x)
ƒ x - 3ƒ
Domain of F(x): [0, ϱ)
f
1x
=
g
ƒ xƒ - 2
Domain: [0, 2)ഫ(2, ϱ)
325
The graphs of y1 = F(x) = 1x ,
y2 = G(x) = ƒ x - 3ƒ, and
F(x)
1x
y3 =
=
are shown
...
Also, the denominator of the quotient
function is equal to zero when x ϭ 3, so we must eliminate this value from the domain
...
F
Composition of Functions
Recall that a function maps every element in the domain to exactly one corresponding
element in the range as shown in the figure on the right
...
Let x correspond to the
original price of each item on the rack
...
Therefore, the function g(x) ϭ 0
...
You have been invited to a special sale that lets you take 10% off the current sale price
and an additional $5 off every item at checkout
...
90g(x) Ϫ 5
determines the checkout price
...
Domain of f
x
g(x) = 0
...
90 g (x) − 5
Domain of g
Range of g
Original price
Sale price 20%
off original price
f(g (x))
Additional 10% off
sale price and $5 off
at checkout
This is an example of a composition of functions, when the output of one function is the
input of another function
...
An algebraic example of this is the function y = 2x2 - 2
...
Recall that the independent variable in function notation is a placeholder
...
Substituting the expression for g(x), we find
f (g(x)) = 2x2 - 2
...
■
Answer:
ƒ x - 3ƒ
G(x)
G
=
a b(x) =
F
F(x)
1x
Domain: (0, ϱ)
x
Domain
f
f(x)
Range
c03c
...
The domain of a composite function is the set of all x such
that g(x) is in the domain of f
...
Therefore, we
restrict the outputs of g(x) Ն 0 and find the corresponding x-values
...
The symbol that represents composition of functions is a small open circle; thus
( f ؠg)(x) = f (g(x)) and is read aloud as “f of g
...
C O M P O S ITI O N
O F F U N CTI O N S
Given two functions f and g, there are two composite functions that can be formed
...
Classroom Example 3
...
3
Let f (x) = 2 1x and
g(x) = -x2 + 4
...
Answer:
( f ؠg) = 2 24 - x2
(g ؠf ) = 4 - 4x
WORDS
DEFINITION
DOMAIN
f composed
with g
ƒ(g(x))
The set of all real numbers x in the
domain of g such that g(x) is also
in the domain of f
...
x
It is important to realize that there are two
“filters” that allow certain values of x into the
domain
...
If x is not in the
domain of g(x), it cannot be in the domain of
( f ؠg)(x) = f(g(x))
...
This adds an
additional filter
...
g(x)
f(g(x))
( f g)(x) = f (g(x))
EXAMPLE 3
Finding a Composite Function
Given the functions ƒ(x) ϭ x2 ϩ 1 and g(x) ϭ x Ϫ 3, find ( f ؠg)(x)
...
f (n) = (n)2 + 1
Express the composite function f ؠg
...
ƒ(g(x)) ϭ (x Ϫ 3)2 ϩ 1
Eliminate the parentheses on the right side
...
c03c
...
4 Operations on Functions and Composition of Functions
Determining the Domain of a Composite Function
EXAMPLE 4
Given the functions f (x) =
1
1
and g(x) = , determine f ؠg, and state its domain
...
f (n) =
1
(n) - 1
Express the composite function f ؠg
...
x
Multiply the right side by
f (g(x)) =
x
...
=
1/x - 1
1 - x
y2 = g(x) =
1
1
- 1
x
1 #x
x
=
x
1
1 - x
- 1
x
x
1 - x
What is the domain of ( f ؠg)(x) ϭ f(g(x))? By inspecting the final result of f(g(x)), we see
that the denominator is zero when x ϭ 1
...
Are there any other values for x that
are not allowed? The function g(x) has the domain x Z 0; therefore we must also exclude zero
...
Domain
of g ؠf is x Z 1, or in interval
notation, ( - ϱ, 1) ഫ (1, ϱ)
...
The domain of the composite function cannot always be determined by examining the
final form of f ؠg
...
Find the domain of ƒ(g(x))
...
EXAMPLE 5
Determining the Domain of a Composite Function
(Without Finding the Composite Function)
▼
CAUTION
The domain of the composite function cannot always be determined by
examining the final form of f ؠg
...
[-3, ϱ)
Find the range of g
...
Since the domain of f is the set of all
real numbers except 2, we eliminate any values of x in the domain of g that correspond
to g(x) ϭ 2
...
Square both sides
...
xϩ3ϭ4
xϭ1
Answer:
Domain f ؠg = [-2, 2]
Domain g ؠf = [0, ϱ)
Classroom Example 3
...
5*
Let f (x) = 11 + 2x and
g(x) = 1 ƒ x ƒ - 1
...
State the domain of ƒ(g(x))
...
4
...
4
...
[-3, 1) ഫ (1, ϱ)
c03c
...
4
...
Calculate, if
possible:
a
...
(g ؠf )(-1)
c
...
g ( f (-3))
EXAMPLE 6
Evaluating a Composite Function
Given the functions ƒ(x) ϭ x2 Ϫ 7 and g(x) ϭ 5 Ϫ x2, evaluate:
a
...
ƒ(g(Ϫ2))
c
...
g(ƒ(Ϫ4))
Solution:
One way of evaluating these composite functions is to calculate the two individual
composites in terms of x: ƒ(g(x)) and g(ƒ(x))
...
Another way of proceeding is as follows:
a
...
Find the value of the inner function g
...
Evaluate ƒ(4)
...
223
b
...
0
d
...
Write the desired quantity
...
Substitute g(Ϫ2) ϭ 1 into f
...
ƒ(g(Ϫ2))
g(Ϫ2) ϭ 5 Ϫ (Ϫ2)2 ϭ 1
ƒ(g(Ϫ2)) ϭ ƒ(1)
ƒ(1) ϭ 12 Ϫ 7 ϭ Ϫ6
f (g(-2)) = - 6
c
...
Find the value of the inner function f
...
Evaluate g(2)
...
Write the desired quantity
...
Substitute ƒ(Ϫ4) ϭ 9 into g
...
g( f(Ϫ4))
f (Ϫ4) ϭ (Ϫ4)2 Ϫ 7 ϭ 9
g( f (Ϫ4)) ϭ g(9)
g(9) ϭ 5 Ϫ 92 ϭ Ϫ76
g( f(-4)) = - 76
■
Answer: ƒ(g(1)) ϭ 5 and
g(ƒ(1)) ϭ Ϫ7
■ YO U R T U R N
Given the functions f (x) ϭ x3 Ϫ 3 and g(x) ϭ 1 ϩ x3, evaluate ƒ(g(1))
and g(ƒ(1))
...
Often, real-world applications are modeled with composite functions
...
The first function maps its input (original
price) to an output (sale price)
...
Example 7 is another real-world application of composite functions
...
This scale is used in
science and is one of the standards of the “metric” (SI) system of measurements
...
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3
...
15ЊC), a hypothetical temperature at which there is a complete
absence of heat energy
...
The symbol for the kelvin is K
...
● With respect to pure water at sea level, the degrees Fahrenheit are gauged by the
spread from 32ЊF (freezing) to 212ЊF (boiling)
...
15
Applications Involving Composite Functions
Classroom Example 3
...
7
Define kelvins as a function of
degrees Fahrenheit
...
Solution:
Degrees Fahrenheit is a function of degrees
Celsius
...
15 into the
equation for F
...
15) + 32
5
Simplify
...
67 + 32
5
F =
Answer:
9
C + 32
5
9
K - 459
...
15
9
SECTION
3
...
The domain of the quotient function is also the intersection
of the domain shared by both f and g with an additional restriction
that g(x) Z 0
...
Rather, the domain of the
composite function is a subset of the domain of g(x)
...
c03c
...
4
■
EXERCISES
SKILLS
f
In Exercises 1–10, given the functions f and g, find f ؉ g, f ؊ g, f
...
g
1
...
f (x) =
2
...
f (x) ϭ 2x2 Ϫ x
7
...
f (x) = 1x - 1
10
...
f (x) =
9
...
f(x) ϭ 3x ϩ 2
g(x) =
1
x
In Exercises 11–20, for the given functions f and g, find the composite functions f ؠg and g ؠf, and state their domains
...
f (x) = 1x - 1
11
...
f(x) ϭ x2 Ϫ 1
g(x) ϭ x2 Ϫ 3
g(x) ϭ 2 Ϫ x
16
...
f (x) = 12 - x
13
...
f (x) = 2x2 - 1
15
...
f (x) =
g(x) =
19
...
f (x) = x2 + 10
21
...
( f ϩ g)(10)
23
...
( f Ϫ g)(5)
25
...
( f ؒ g)(5)
f
27
...
a b (2)
g
29
...
ƒ(g(1))
31
...
g(ƒ(4))
33
...
g(ƒ(0))
35
...
( f ؠg)(4)
38
...
g A f A 17B B
In Exercises 39–50, evaluate f( g(1)) and g( f(2)), if possible
...
f (x) = 13 - x,
39
...
f (x) = 1x - 1,
48
...
f (x) = x2 + 1,
3
46
...
f (x) =
1
,
ƒ x - 1ƒ
49
...
f (x) = 11 - x,
44
...
f (x) =
g(x) = x2 + 2
g(x) = 1x - 3
1
,
x2 - 3
g(x) = ƒ2x - 3 ƒ
1/2
50
...
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Page 331
3
...
f (x) = 1x - 1,
In Exercises 51–60, show that f (g(x)) ؍x and g( f(x)) ؍x
...
f (x) = 2x + 1, g(x) =
2
g(x) = x2 + 1 for x Ն 1
1
for x Z 0
x
1x + 9
57
...
f (x) =
x
x - 1
55
...
f (x) =
x - 2
,
3
331
g(x) = 12 - x for x Յ 2
g(x) = 3x + 2
54
...
f (x) = 18x - 1,
56
...
f (x) = 225 - x2,
g(x) ϭ 5 Ϫ x3
g(x) = 225 - x2 for 0 … x … 5
g(x) =
x3 + 1
8
In Exercises 61–66, write the function as a composite of two functions f and g
...
)
61
...
f (g(x)) =
1x + 1 - 2
63
...
f (g(x)) = 21 - x2
62
...
f (g(x)) =
1
1 + x2
1x
31x + 2
A P P L I C AT I O N S
Exercises 67 and 68 depend on the relationship between
degrees Fahrenheit, degrees Celsius, and kelvins:
9
F = C + 32
C = K - 273
...
Temperature
...
68
...
Convert the following degrees Fahrenheit to
kelvins: 32ЊF and 212ЊF
...
Dog Run
...
Fencing is purchased in linear feet
...
Write a composite function that determines the area of your
dog pen as a function of how many linear feet are purchased
...
If you purchase 100 linear feet, what is the area of your
dog pen?
c
...
Dog Run
...
Fencing is purchased in
linear feet
...
Write a composite function that determines the area of
your dog pen as a function of how many linear feet are
purchased
...
If you purchase 100 linear feet, what is the area of your
dog pen?
c
...
Market Price
...
Assume that the market price p and the number of
units for sale x are related by the demand equation:
1
x
2
Assume that the cost C(x) of producing x items is governed
by the equation
C(x) = 2000 + 10x
p = 3000 -
and the revenue R(x) generated by selling x units is governed by
R(x) = 100x
a
...
b
...
c
...
72
...
Typical supply and demand relationships state
that as the number of units for sale increases, the market price
decreases
...
Write the cost as a function of price p
...
Write the revenue as a function of price p
...
Write the profit as a function of price p
...
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C H A P T E R 3 Functions and Their Graphs
In Exercises 73 and 74, refer to the following:
75
...
If the radius of the oil spill is given by
r(t) = 10t - 0
...
The number of
products manufactured n is a function of the number of hours t the
assembly line is operating and is given by the function n(t)
...
73
...
If the quantity of a product manufactured during a
day is given by
n(t) = 50t - t2
and the area of the oil spill is given by
A(r) = pr2
a
...
b
...
76
...
If the radius of the oil spill is given by
r(t) = 8t - 0
...
Find a function that gives the cost of manufacturing the
product in terms of the number of hours t the assembly
line was functioning, C(n(t))
...
Find the cost of production on a day when the assembly
line was running for 16 hours
...
74
...
If the quantity of a product manufactured during a
day is given by
n(t) = 100t - 4t2
and the cost of manufacturing the product is given by
C(n) = 8n + 2375
a
...
b
...
Interpret your answer
...
Find a function that gives the area of the oil spill in terms
of the number of days since the start of the spill, A(r(t))
...
Find the area of the oil spill to the nearest square mile 5
days after the start of the spill
...
Environment: Oil Spill
...
Find the area of the spill as a
function of time
...
Pool Volume
...
If 50 cubic feet of water is pumped into the
pool per hour, write the water-level height (feet) as a
function of time (hours)
...
Fireworks
...
If the
family is 2 miles from where the fireworks are being
launched and the fireworks travel vertically, what is the
distance between the family and the fireworks as a function
of height above ground?
Surveys performed immediately following an accidental oil spill
at sea indicate the oil moved outward from the source of the spill
in a nearly circular pattern
...
80
...
A couple are about to put their house up
for sale
...
Write a function that represents the
amount of money they will make on their home as a
function of the asking price p
...
Explain
the mistake that is made in each problem
...
g
f
Solution:
82
...
What mistake was made?
This is incorrect
...
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Page 333
3
...
f ؠg
85
...
Given the function ƒ(x) ϭ x2 ϩ 7 and g(x) = 1x - 3, find
f ؠg, and state the domain
...
What mistake was made?
This is incorrect
...
f (x) - g(x) = x + 2 - x2 - 4
f ؠg = f (g(x)) = A 1x - 3 2 + 7
= - x2 + x - 2
Domain: (Ϫϱ, ϱ)
This is incorrect
...
What mistake was made?
■
CONCEPTUAL
In Exercises 87–90, determine whether each statement is true or false
...
When adding, subtracting, multiplying, or dividing two
functions, the domain of the resulting function is the union of
the domains of the individual functions
...
For any functions f and g, ( f ؠg)(x) exists for all values of x
that are in the domain of g(x), provided the range of g is a
subset of the domain of f
...
For any functions f and g, ƒ(g(x)) ϭ g(ƒ(x)) for all values of x
that are in the domain of both f and g
...
The domain of a composite function can be found by
inspection, without knowledge of the domain of the
individual functions
...
For the functions ƒ(x) ϭ x ϩ a and g(x) =
, find
x - a
g ؠf and state its domain
...
For the functions ƒ(x) ϭ ax2 ϩ bx ϩ c and g(x) =
,
x - c
find g ؠf and state its domain
...
Using a graphing utility, plot y1 = 1x + 7 and y2 = 19 - x
...
Using a graphing utility, plot y1 = 1x + 5, y2 =
,
13 - x
y1
and y3 =
...
What is the domain of y3?
93
...
1
1
94
...
Assume a 7 1 and b 7 1
...
Using a graphing utility, plot y1 = 2x2 - 3x - 4,
1
1
y2 = 2
, and y3 = 2
...
The graph of y3 is only defined
for the domain of g ؠf
...
98
...
If y1 represents a function f and y2 represents a
function g, then y3 represents the composite function g ؠf
...
State the
domain of g ؠf
...
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SECTION
3
...
Verify that two functions are inverses of one another
...
Find the inverse of a function
...
Understand why functions and their inverses are
symmetric about y ϭ x
...
These
are examples of functions, where a person is the input and the output is blood type or
DNA sequence
...
The difference between these functions
is that many people have the same blood type, but DNA is unique to each individual
...
When a function has a one-to-one correspondence, like
the DNA example, then mapping backwards is possible
...
Determine Whether a Function Is One-to-One
In Section 3
...
Algebraically, each value for x can
correspond to only a single value for y
...
3
...
Although the square
function and the absolute value function map each value of x to exactly one value for y,
these two functions map two values of x to the same value for y
...
The identity and reciprocal functions, on the other hand, map each
x to a single value for y, and no two x-values map to the same y-value
...
DEFINITION
One-to-One Function
A function ƒ(x) is one-to-one if no two elements in the domain correspond to the
same element in the range; that is,
if x1 Z x2, then f(x1) Z f(x2)
...
334
c03c
...
5 One-to-One Functions and Inverse Functions
EXAMPLE 1
Determining Whether a Function Defined as a Set of
Points Is a One-to-One Function
For each of the three relations, determine whether the relation is a function
...
f = {(0, 0), (1, 1), (1, -1)}
g = {(-1, 1), (0, 0), (1, 1)}
Classroom Example 3
...
1
Determine whether these
functions are one-to-one
functions
...
f ϭ {(Ϫ2, Ϫ2), (Ϫ1, Ϫ1),
(0, 0), (1, Ϫ1)}
b
...
no
h = {(-1, -1), (0, 0), (1, 1)}
b
...
g is a function,but not
one-to-one
...
Just as there is a graphical test for functions, the vertical line test, there is a graphical
test for one-to-one functions, the horizontal line test
...
The identity and reciprocal functions, however, will intersect
a horizontal line in at most only one point
...
DEFINITION
Horizontal Line Test
If every horizontal line intersects the graph of a function in at most one point, then
the function is classified as a one-to-one function
...
If it is a function,
determine whether it is a one-to-one function
...
x = y2
y = x2
y = x3
335
Classroom Example 3
...
2
Determine whether these
functions are one-to-one
functions
...
y = ƒ x ƒ - 1
3
b
...
no b
...
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Page 336
C H A P T E R 3 Functions and Their Graphs
Solution:
x = y2
y = x2
y
y = x3
y
y
5
10
10
x
x
10
–5
5
x
–5
–5
–10
5
Not a function
Answer:
a
...
no
One-to-one function
(fails vertical line test)
■
Function, but not
one-to-one
(passes vertical line test
but fails horizontal line test)
(passes both horizontal
and vertical line tests)
■ YO U R T U R N
Determine whether each of the functions is a one-to-one function
...
ƒ(x) ϭ x ϩ 2
b
...
In the Your Turn following Example 2, we found (using the horizontal line test) that
ƒ(x) ϭ x ϩ 2 is a one-to-one function, but that ƒ(x) ϭ x2 ϩ 1 is not a one-to-one
function
...
W OR DS
M ATH
State the function
...
Subtract 2 from both sides of the equation
...
W OR DS
M ATH
State the function
...
f(x) ϭ x2 ϩ 1
Subtract 1 from both sides of the equation
...
x2 ϭ x2
1
2
x1 ϭ ± x2
x2 ϩ 1 ϭ x2 ϩ 1
1
2
ƒ(x) ϭ x2 ϩ 2 is not a one-to-one function
...
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Page 337
3
...
ƒ(x) ϭ 5x3 Ϫ 2
Classroom Example 3
...
3
Determine whether these are
one-to-one algebraically
...
ƒ(x) ϭ ƒ x ϩ 1 ƒ
Solution (a):
ƒ(x1) ϭ 5x3 Ϫ 2 and f(x2) ϭ 5x3 Ϫ 2
1
2
Find ƒ(x1) and f(x2)
...
5x3 Ϫ 2 ϭ 5x3 Ϫ 2
1
2
5x3 ϭ 5x3
1
2
Add 2 to both sides of the equation
...
1/3
1x32
1
Take the cube root of both sides of the equation
...
y = -312x + 1
b
...
* y = (x - 2)3 - 2
Answer:
a
...
no
c
...
f(x) ϭ 5x Ϫ 2 is a one-to-one function
...
Let ƒ(x1) ϭ ƒ(x2)
...
(x1 ϩ 1) ϭ (x2 ϩ 1) or (x1 ϩ 1) ϭ Ϫ(x2 ϩ 1)
x1 ϭ x2 or x1 ϭ Ϫx2 Ϫ 2
ƒ(x) ϭ ƒ x ϩ 1 ƒ is not a one-to-one function
...
This implies that there is a one-to-one correspondence
between the inputs (domain) and outputs (range) of a one-to-one function f (x)
...
The function that maps the
output back to the input of a function f is called the inverse function and is denoted f Ϫ1(x)
...
Therefore, the inverse function f Ϫ1 maps every y back to a
unique and distinct x
...
For example, let the function h(x) ϭ {(Ϫ1, 0), (1, 2), (3, 4)}
...
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C H A P T E R 3 Functions and Their Graphs
Domain
of f
x
Range
of f
f (x) = 5x
5x
?
x
f (x) = 5x
5x
f ؊1(5x) = x
▼
Page 338
The inverse function undoes whatever the function does
...
If we want to map
backwards or undo the 5x, we develop a function called the inverse function that takes 5x
as input and maps back to x as output
...
Note that if we
5
input 5x into the inverse function, the output is x: f -1(5x) = 1(5x) = x
...
The function g is denoted by ƒϪ1 (read
“f-inverse”)
...
The Ϫ1 is not used as an exponent and, therefore,
1
does not represent the reciprocal of f :
...
Domain of ƒ ϭ range of ƒ Ϫ1 and range of ƒ ϭ domain of ƒϪ1
ƒϪ1(ƒ(x)) ϭ x and
Classroom Example 3
...
4
Verify that
f -1(x) = (x - 1)3 + 2 is the
inverse of
3
f (x) = 1 x - 2 + 1
...
EXAMPLE 4
Verify that f (x) =
-1
ƒ(ƒϪ1(x)) ϭ x
Verifying Inverse Functions
1
2x
- 2 is the inverse of ƒ(x) ϭ 2x ϩ 4
...
Write ƒϪ1 using placeholder notation
...
Simplify
...
Substitute f ؊1(x) 1 ؍x ؊ 2 into f
...
f (n) (2 ؍n) ؉ 4
1
f ( f ؊1(x)) 2 ؍a x ؊ 2b ؉ 4
2
ƒ(ƒ ؊1(x)) ؍x ؊ 4 ؉ 4 ؍x
ƒ(ƒ ؊1(x)) ؍x
c03c
...
5 One-to-One Functions and Inverse Functions
339
Note the relationship between the domain and range of f and ƒϪ1
...
Verifying Inverse Functions with Domain Restrictions
EXAMPLE 5
Ϫ1
2
Substitute f(x) = 1x into ƒ
...
Ϫ1
Write ƒ
Classroom Example 3
...
5*
Verify that
is the inverse of
f (x) = 3 + 12x
...
Ϫ1
f
؊1
2
f(n) (1 ؍n)
ƒ(ƒ (x)) 1 ؍x2 ؍x, x » 0
ƒ؊1(ƒ(x)) ؍x for x » 0
Write f using placeholder notation
...
2
f (x) = 1x
ƒ(ƒ؊1(x)) ؍x for x » 0
DOMAIN
[0, ϱ)
[0, ϱ)
[0, ϱ)
ƒϪ1(x) ϭ x2, x Ն 0
RANGE
[0, ϱ)
Graphical Interpretation of Inverse Functions
In Example 4, we showed that f ؊1(x) 1 ؍x ؊ 2 is the inverse of ƒ(x) 2 ؍x ؉ 4
...
ƒ؊1(x)
ƒ(x)
x
y
x
؊3
؊2
؊2
؊3
؊2
0
0
؊2
؊1
2
2
؊1
0
4
4
y
y
0
(0, 4)
(–1, 2)
(–2, 0)
(–3, –2)
(4, 0)
x
(2, –1)
(0, –2)
(–2, –3)
Note that the point (؊3, ؊2) lies on the function and the point (؊2, ؊3) lies on the inverse
...
Draw the line y ϭ x on the graph
...
In general, if the point (a, b) is on the graph of a function, then the point (b, a) is on the
graph of its inverse
...
Notice the interchanging of the
x- and y-coordinates
...
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Page 340
C H A P T E R 3 Functions and Their Graphs
EXAMPLE 6
Graphing the Inverse Function
y
Given the graph of the function f (x),
plot the graph of its inverse f Ϫ1(x)
...
(2, 4)
(0, 2)
(4, 2)
x
(–2, 0)
(2, 0)
(–3, –2)
(0, –2)
(–2, –3)
■
Answer:
■ YO U R T U R N
y
Given the graph of a function
f, plot the inverse function
...
At this point, you should be able to determine whether two functions are inverses of one another
...
The symmetry about the line y ϭ x tells us that the roles of x and y
interchange
...
Algebraically, this corresponds to
interchanging x and y
...
Earlier, we found that if h(x) ϭ {(Ϫ1, 0), (1, 2), (3, 4)},
then hϪ1(x) ϭ {(0, Ϫ1), (2, 1), (4, 3)}
...
This relationship implies that
ƒ(x) ؍y and ƒ؊1( y) ؍x
...
Now consider the
c03c
...
5 One-to-One Functions and Inverse Functions
function defined by ƒ(x) ϭ 3x Ϫ 1
...
Solve for the variable x:x = 1 y + 1
...
It is
3
3
customary to write the independent variable as x, so we write the inverse as
f -1( x) = 1 x + 1
...
5
...
a
...
1
1
f 1 f -1(x)2 = 3a x + b - 1 = x + 1 - 1 = x
3
3
1
1
1
1
f -1 ( f (x)) = (3x - 1) + = x - + = x
3
3
3
3
FINDING
TH E I NVE R S E O F A F U N CTI O N
b
...
Then the following procedure can be used to find the
inverse function f Ϫ1 if the inverse exists
...
2
Solve the resulting equation for x in
terms of y (if possible)
...
f -1( y) = - 1 y +
3
5
3
4
Let y ϭ x (interchange x and y)
...
)
Answer:
a
...
STEP
PROCEDURE
1
Let y ϭ ƒ(x)
...
3
Solve for y in terms of x
...
Note the following:
■
■
■
Verify first that a function is one-to-one prior to finding an inverse (if it is not
one-to-one, then the inverse does not exist)
...
The domain of f is the range
of ƒϪ1 and vice versa
...
(There is a closed hole at (2, 0)
and an open hole at (4, 0)
...
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Page 342
Find the inverse of the function f(x) = 1x + 2
...
C H A P T E R 3 Functions and Their Graphs
Using a graphing utility, plot
y1 = f (x) = 1x + 2,
Technology Tip
y2 ϭ ƒϪ1(x) ϭ x2 Ϫ 2, and
y3 ϭ x
...
x
–5
5
y = 1x + 2
Note that the function f(x) and its
inverse ƒϪ1(x) are symmetric about
the line y ϭ x
...
S TEP 2
Interchange x and y
...
Square both sides of the equation
...
S TEP 4
Study Tip
Had we ignored the domain and
range in Example 7, we would have
found the inverse function to be the
square function f(x) ϭ x2 Ϫ 2, which
is not a one-to-one function
...
Classroom Example 3
...
7
Determine the inverse of the
following functions and state
the domain and range
...
f (x) = 11 - 3x
b
...
g-1(x) = (1 - x)2 - 2
Range: [-2, ϱ)
Domain: (- ϱ, 1]
a
...
ƒϪ1(x) ϭ x2 Ϫ 2
Note any domain restrictions
...
)
f:
Domain: [Ϫ2, ϱ)
Range: [0, ϱ)
The inverse of f (x) = 1x + 2 is f (x) = x - 2 for x Ú 0
...
ƒϪ1(ƒ(x)) ϭ x for all x in the
domain of f
...
Note that the function f(x) = 1x + 2
( f (x)) = A1x + 2B - 2
= x + 2 - 2 for x Ú -2
= x
f ( f -1(x)) = 21x2 - 22 + 2
f
2
-1
= 2x2 for x Ú 0
= x
y
f –1(x)
5
and its inverse ƒϪ1(x) ϭ x2 Ϫ 2 for
x Ն 0 are symmetric about the line y ϭ x
...
f -1(x) =
b
...
g(x) = 1x - 1
Find the inverse of the given function
...
a
...
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Page 343
3
...
Classroom Example 3
...
8
Determine whether the inverse
exists
...
f (x) = 1 - ƒ xƒ
y
f (x) = | x |
Solution:
The function ƒ(x) ϭ ƒ x ƒ fails the horizontal
line test and therefore is not a one-to-one
function
...
EXAMPLE 9
343
b
...
no
x
Finding the Inverse Function
1
2 (1
+ x)2
b
...
Find its inverse
...
y =
2
x + 3
S TEP 2
Interchange x and y
...
Multiply the equation by (y ϩ 3)
...
xy ϩ 3x ϭ 2
Subtract 3x from both sides
...
The graphs of y1 = f (x) =
Divide the equation by x
...
f-1(x) = -3 +
Note any domain restrictions on f ؊1(x)
...
2
x
x Z 0
2
2
, x Z -3, is f -1(x) = -3 + , x Z 0
...
f -1( f (x)) = -3 +
f 1 f -1(x)2 =
■ YO U R T U R N
2
= -3 + (x + 3) = x, x Z -3
2
b
a
x + 3
2
2
=
= x, x Z 0
2
2
a-3 + b + 3
a b
x
x
The function f (x) =
4
, x Z 1, is a one-to-one function
...
Note in Example 9 that the domain of f is (- ϱ , -3)ഫ(-3, ϱ) and the domain of f Ϫ1 is
(- ϱ, 0) ഫ (0, ϱ)
...
Study Tip
The range of the function is equal to
the domain of its inverse function
...
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C H A P T E R 3 Functions and Their Graphs
Finding the Inverse of a Piecewise-Defined Function
EXAMPLE 10
The function f (x) = b
3x
x2
x 6 0
, is a one-to-one function
...
x Ú 0
y
Solution:
25
From the graph of ƒ we can make a table
with corresponding domain and range values
...
DOMAIN
OF
f / RANGE
OF
f Ϫ1
RANGE
OF
f / DOMAIN
(-ϱ, 0)
f Ϫ1
(-ϱ, 0)
[0, ϱ)
OF
[0, ϱ)
f (x) = 3x on (- ϱ, 0); find ƒϪ1(x) on (- ϱ, 0)
...
y ϭ 3x
S TEP 2
Solve for x in terms of y
...
y = 1x
3
S TEP 4
Let y ϭ ƒϪ1(x)
...
S TEP 1
S TEP 2
Solve for x in terms of y
...
S TEP 4
Let y ϭ ƒϪ1(x)
...
y ϭ x2
The range of ƒϪ1 is [0, ϱ)
f-1(x) = ; 1x
x ϭ y2
f-1(x) = 1x
1
x
3
f -1(x) = c
1x
Combining the two pieces yields a piecewise-defined inverse function
...
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3
...
5
S U M MARY
One-to-One Functions
Each input in the domain corresponds to exactly one output in the
range, and no two inputs map to the same output
...
1
...
2
...
3
...
Properties of Inverse Functions
1
...
2
...
Composition of inverse functions
■ ƒϪ1(ƒ(x)) ϭ x for all x in the domain of f
...
4
...
Procedure for Finding the Inverse of a Function
1
...
3
...
Let y ϭ ƒ(x)
...
Solve for y
...
SECTION
3
...
If it is a function, determine whether it is
a one-to-one function
...
Domain
Range
MONTH
October
January
April
2
...
4
...
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C H A P T E R 3 Functions and Their Graphs
6
...
{(0, 1), (1, 2), (2, 3), (3, 4)}
7
...
{(0, 1), (1, 1), (2, 1), (3, 1)}
10
...
{(0, 1), (1, 0), (2, 1), (Ϫ2, 1), (5, 4), (Ϫ3, 4)}
11
...
y
13
...
(0, 1)
x
x
x
(2, –2)
15
...
y
10
y
5
(–2, 3)
x
(2, –1)
x
–5
x
5
10
In Exercises 17–24, determine algebraically and graphically whether the function is one-to-one
...
ƒ(x) ϭ ƒ x Ϫ 3 ƒ
18
...
f (x) =
20
...
ƒ(x) ϭ x2 Ϫ 4
22
...
f(x) ϭ x3 Ϫ 1
24
...
Graph
f(x) and f ؊1(x) on the same axes to show the symmetry about the line y ؍x
...
f (x) = 1x - 1, x Ú 1; f -1(x) = x2 + 1, x Ú 0
25
...
f (x) = 2 - x2, x Ú 0; f -1(x) = 12 - x, x … 2
26
...
f (x) =
1 -1
1
; f (x) = , x Z 0
x
x
30
...
f (x) =
1
1
, x Z -3; f -1(x) =
- 3, x Z 0
2x + 6
2x
32
...
f (x) =
x + 3
3 - 4x
, x Z -4; f -1(x) =
,x Z 1
x + 4
x - 1
34
...
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3
...
35
...
37
...
y
y
y
10
5
(–3, 3)
(3, 1)
x
(1, 1)
x
x
10
(–1, –3)
x
–2
39
...
41
...
y
y
5
8
y
x
10
10
5
x
–5
x
–8
2
5
x
–10
5
–10
In Exercises 43–60, the function f is one-to-one
...
State the domain and range of
both f and f ؊1
...
ƒ(x) ϭ x Ϫ 1
44
...
ƒ(x) ϭ x3 ϩ 1
49
...
ƒ(x) ϭ x3 Ϫ 1
51
...
ƒ(x) ϭ 2x2 ϩ 1, x Ն 0
53
...
ƒ(x) ϭ (x Ϫ 3)2 Ϫ 2, x Ն 3
50
...
ƒ(x) ϭ Ϫ3x ϩ 2
55
...
59
...
ƒ(x) ϭ 2x ϩ 3
3
x
7
58
...
f (x) =
7 + x
2
x
56
...
If it is a one-to-one
function, find its inverse
...
G(x) = e
0
1x
x 6 0
x Ú 0
1
62
...
ƒ(x) = u x3
x
x … -1
-1 6 x 6 1
x Ú 1
x + 3
64
...
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348
■
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Page 348
C H A P T E R 3 Functions and Their Graphs
A P P L I C AT I O N S
Security, write a function E(x) that expresses the student’s
take-home pay each week
...
What does the inverse function tell you?
65
...
The equation used to convert from degrees
Celsius to degrees Fahrenheit is f (x) = 9 x + 32
...
What does the
inverse function represent?
66
...
The equation used to convert from degrees
Fahrenheit to degrees Celsius is C(x) = 5 (x - 32)
...
What does the inverse
function represent?
67
...
The Richmond rowing club is planning to enter
the Head of the Charles race in Boston and is trying to
figure out how much money to raise
...
Find the cost function C(x) as a function of the
number of boats the club enters x
...
68
...
A phone company charges
$
...
12 per minute every minute after that
...
Suppose you buy a “prepaid”
phone card that is planned for a single call
...
In Exercises 71–74, refer to the following:
By analyzing available empirical data it was determined that
during an illness a patient’s body temperature fluctuated during
one 24-hour period according to the function
T(t) = 0
...
70
where T represents that patient’s temperature in degrees
Fahrenheit and t represents the time of day in hours measured
from 12:00 A
...
(midnight)
...
Health/Medicine
...
72
...
Find time as a function of temperature,
that is, the inverse function t(T)
...
Health/Medicine
...
74
...
At what time, to the nearest hour, was the
patient’s temperature 99
...
Salary
...
If
Target withholds 25% of his earnings for taxes and Social
■
70
...
A grocery store pays you $8 per hour for the first
40 hours per week and time and a half for overtime
...
Find the
inverse function EϪ1(x)
...
A linear one-to-one function is graphed below
...
In Exercises 75–78, explain the mistake that is made
...
Is x ϭ y a one-to-one
function?
2
y
Solution:
Solution:
Yes, this graph
represents a one-to-one
function because it
passes the horizontal
line test
...
What mistake was made?
x
Note that the points
(3, 3) and (0, ؊4) lie
on the graph of the
function
...
(0, 4)
(3, 3)
x
(–3, –3)
(0, –4)
This is incorrect
...
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Page 349
78
...
3
...
Given the function ƒ(x) ϭ x2, find the inverse function ƒϪ1(x)
...
y = 1x
y ϭ x2
Step 2: Solve for x
...
x = 1y - 2
Step 3: Solve for y
...
ƒ -1(x) = 1x
ƒϪ1(x) ϭ x2 ϩ 2
Step 1: Let y ϭ ƒ(x)
...
Step 4: Let y ϭ ƒϪ1(x)
...
Step 3: Interchange x and y
...
domain restriction that x Ն 2
...
What mistake was made?
The domain of ƒϪ1(x) is x Ն 2
...
What mistake was made?
■
CONCEPTUAL
In Exercises 79–82, determine whether each statement is true or false
...
Every even function is a one-to-one function
...
Every odd function is a one-to-one function
...
If (0, b) is the y-intercept of a one-to-one function ƒ, what is
the x-intercept of the inverse ƒϪ1?
84
...
It is not possible that ƒ ϭ ƒϪ1
...
A function ƒ has an inverse
...
■
CHALLENGE
85
...
If we restrict ourselves to
the semicircle that lies in quadrants I and II, the graph represents a function, but it is not a one-to-one function
...
Determine
the equations of both the one-to-one function and its inverse
...
■
c
86
...
x
87
...
Assuming that the conditions found in Exercise 87 are met,
determine the inverse of the linear function
...
89
...
ƒ(x) ϭ x1/3 Ϫ x5
90
...
ƒ(x) =
3
x3 + 2
1
x1͞2
In Exercises 93–96, graph the functions f and g and the line
y ؍x in the same screen
...
f (x) = 13x - 5;
94
...
f (x) = 1x + 3 - 2;
g(x) =
x2
5
+
3
3
4
x2
- ,x Ú 0
3
3
95
...
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Page 350
SECTION
3
...
Develop mathematical models using inverse variation
...
Develop mathematical models using joint variation
...
Understand the difference between combined variation
and joint variation
...
Two quantities in
the real world often vary with respect to one another
...
For
example, the more money we make, the more total dollars of federal income tax we expect
to pay
...
For example, when interest rates on
mortgages decrease, we expect the number of homes purchased to increase because a buyer
can afford “more house” with the same mortgage payment when rates are lower
...
Direct Variation
When one quantity is a constant multiple of another quantity, we say that the quantities are
directly proportional to one another
...
The following are equivalent statements:
■
■
■
y ϭ kx, where k is a nonzero constant
...
y is directly proportional to x
...
In 2005, the national average cost of residential electricity was 9
...
For example, if a residence used 3400 kWh, then the bill would be $324
...
25
...
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3
...
y ϭ kx
Classroom Example 3
...
1
Find a mathematical model
describing the monthly
long-distance bill if the cost is
directly proportional to the
number of minutes, and a
customer that used 160 minutes
is billed $40
...
x ϭ number of kWh
y ϭ cost (dollars)
k ϭ cost per kWh
Answer: y = 1 x, where
4
x ϭ number of minutes and
y ϭ cost (in dollars)
...
If a household in Tennessee on average used 3098 kWh per
month and had an average monthly electric bill of $179
...
Solution:
Substitute the given data x ϭ 3098 kWh
and y ϭ $179
...
179
...
k =
179
...
05810
3098
y = 0
...
81 ¢/kWh
...
Not all variation we see in nature is direct variation
...
If organisms grew isometrically, young children would look just like adults, only
smaller
...
The relative proportions of a human body
change dramatically as the human grows
...
Allometric growth is the pattern of growth whereby different parts
of the body grow at different rates with respect to each other
...
D I R E CT
VAR IATI O N WITH P O WE R S
Let x and y represent two quantities
...
y varies directly with the nth power of x
...
One example of direct variation with powers is height and weight of humans
...
W = kH3
■
Answer: y ϭ 0
...
c03d
...
6
...
Answer: W =
165 3
H
(5
...
Must be athletic, smart, like the movies and dogs, and have height and weight
similarly proportioned to mine
...
How much would a 5Ј6Љ woman weigh who has the same proportionality as the male?
Solution:
Write the direct variation (cube)
model for height versus weight
...
Solve for k
...
898148 L 0
...
9H 3
Let H ϭ 5
...
W ϭ 0
...
5)3 ϭ 149
...
■
Answer: 200 pounds
■ YO U R T U R N
A brother and sister both have weight (pounds) that varies as the cube
of height (feet) and they share the same proportionality constant
...
Her brother is 6 feet 4
inches
...
Supply is the quantity
that producers are willing to sell at a given price
...
Demand is the quantity of a good that consumers are
not only willing to purchase but also have the capacity to buy at a given price
...
There may be 1 billion people who want
to buy the filet mignon but don’t have the financial means to do so
...
Demand can be modeled with an inverse variation of price: when the price increases,
demand decreases, and vice versa
...
The following are equivalent statements:
k
■ y =
, where k is a nonzero constant
...
■ y is inversely proportional to x
...
11/24/11
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Page 353
3
...
If the number of potential buyers of a house in a
particular city is inversely proportional to the price of
the house, find a mathematical equation that describes
the demand for houses as it relates to price
...
qxd
(100, 1000)
800
600
400
200
(200, 500)
(400, 250)
(600, 167)
200 400 600 800
Price of the house
(in thousands of dollars)
Solution:
Write the inverse variation model
...
k
x
x ϭ price of house in thousands of dollars
y ϭ number of buyers
y =
Select any point that lies
on the curve
...
x
Solve for k
...
100,000
x
y =
100,000
= 50
2000
There are only 50 potential buyers for a $2 million house in this city
...
If there are
12,500 potential buyers for a $2 million condominium, how many
potential buyers are there for a $5 million condominium?
Two quantities can vary inversely with the nth power of x
...
If x and y are related by the equation y =
Joint Variation and Combined Variation
We now discuss combinations of variations
...
When direct
variation and inverse variation occur at the same time, the variation is called combined
variation
...
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C H A P T E R 3 Functions and Their Graphs
An example of a joint variation is simple interest (Section 1
...
Note that if the interest rate increases, then the interest earned also increases
...
An example of combined variation is the combined gas law in chemistry,
P = k
where
■
■
■
■
T
V
P is pressure
T is temperature (kelvins)
V is volume
k is a gas constant
This relation states that the pressure of a gas is directly proportional to the temperature
and inversely proportional to the volume containing the gas
...
As an example, the gas in the headspace of a soda bottle has a fixed volume
...
Compare the different pressures of
opening a twist-off cap on a bottle of soda that is cold versus one that is hot
...
”
EXAMPLE 4
Classroom Example 3
...
4*
Write an equation describing
the following situation: E is
directly proportional to m and
the square of c
...
1803
when m ϭ 3 and c ϭ 3
...
Answer: E = mc2
Combined Variation
The gas in the headspace of a soda bottle has a volume of 9
...
If the
soda bottle is stored in a refrigerator, the temperature drops to approximately 279 K
(42ЊF)
...
Let P ϭ 2 atm, T ϭ 298 K, and V ϭ 9
...
Solve for k
...
0 in P = k
...
T
V
298
2 = k
9
18
k =
298
18 # 279
P =
L 1
...
87 atm
c03d
...
6 Modeling Functions Using Variation
355
SECTION
3
...
For two quantities
x and y, we say that
■
■
y is directly proportional to x if y 5 kx
...
x
Joint variation occurs when one quantity is directly proportional
to two or more quantities
...
SECTION
3
...
Use k as the constant of variation
...
z varies directly with m
...
h varies directly with 1t
...
ƒ varies inversely with
...
P varies inversely with r2
...
y varies directly with x
...
s varies directly with t
...
A varies directly with x2
...
V varies directly with x
...
F varies directly with w and inversely with L
...
V varies directly with T and inversely with P
...
v varies directly with both g and t
...
S varies directly with both t and d
...
R varies inversely with both P and T
...
y varies inversely with both x and z
...
y is directly proportional to the square root of x
...
y is inversely proportional to the cube of t
...
17
...
d ϭ r when t ϭ 1
...
F is directly proportional to m
...
19
...
V ϭ 6h when w ϭ 3 and l ϭ 2
...
A is directly proportional to both b and h
...
21
...
A ϭ 9p when r ϭ 3
...
V varies directly with the cube of r
...
4
...
W is directly proportional to both R and the square of I
...
25
...
V varies directly with both h and r2
...
V varies inversely with P
...
26
...
I ϭ 42 when d ϭ 16
...
F varies inversely with both l and L
...
c03d
...
y varies inversely with both x and z
...
05
...
t varies inversely with s
...
4 when s ϭ 8
...
W varies inversely with the square of d
...
2
...
R varies inversely with the square of I
...
4 when I ϭ 3
...
32
...
y ϭ 12 when x ϭ 0
...
33
...
R ϭ 0
...
4
...
F varies directly with m and inversely with d
...
35
...
F ϭ 20 when m1 ϭ 8, m2 ϭ 16, and d ϭ 0
...
36
...
w ϭ 20 when g ϭ 16 and t ϭ 0
...
■
A P P L I C AT I O N S
37
...
Jason and Valerie both work at Panera Bread and
have the following paycheck information for a certain week
...
E MPLOYEE
HOURS WORKED
WAGES
Jason
23
$172
...
00
Exercises 41 and 42 are examples of the golden ratio, or phi, a
proportionality constant that appears in nature
...
618
...
goldenratio
...
41
...
The length of your forearm F (wrist
to elbow) is directly proportional to the length of your
hand H (length from wrist to tip of middle finger)
...
8 inches
...
38
...
The sales tax in Orange and Seminole counties in
Florida differs by only 0
...
A new resident knows this but
doesn’t know which of the counties has the higher tax
...
If the tax on a pair of $40
sneakers is $2
...
84 in Seminole County, write two equations: one
for each county that describes the tax T, which is directly
proportional to the purchase price P
...
Aircraft traveling at a subsonic speed (less than
the speed of sound) have a Mach number less than 1
...
Aircraft traveling at a supersonic speed (greater than the
speed of sound) have a Mach number greater than 1
...
39
...
The U
...
Navy Blue Angels fly F-18 Hornets that
are capable of Mach 1
...
How fast can F-18 Hornets fly at sea
level?
40
...
The U
...
Air Force’s newest fighter aircraft is the
F-35, which is capable of Mach 1
...
How fast can an F-35 fly
at sea level?
42
...
Each section of your index finger, from the
tip to the base of the wrist, is larger than the preceding one by
about the golden (Fibonacci) ratio
...
1
2
2
3
4
3
5
6
7
8
5
9
10
11
12
13
14
15
8
16
17
18
c03d
...
6 Modeling Functions Using Variation
357
For Exercises 43 and 44, refer to the following:
For Exercises 49 and 50, refer to the following:
Hooke’s law in physics states that if a spring at rest (equilibrium
position) has a weight attached to it, then the distance the spring
stretches is directly proportional to the force (weight), according
to the formula:
F = kx
In physics, the inverse square law states that any physical quantity
or strength is inversely proportional to the square of the distance
from the source of that physical quantity
...
Below
is a table of average distances from the Sun:
where F is the force in Newtons (N), x is the distance stretched in
meters (m), and k is the spring constant (N/m)
...
Physics
...
How far will a force of 72 N stretch the spring?
44
...
A force of 30 N will stretch the spring
10 centimeters
...
Business
...
If the company charges $17
...
Economics
...
Demand for the product is
10,000 units when the price is $5
...
Find the
demand for the product (to the nearest hundred units) when
the price is $6
...
47
...
Levi’s makes jeans in a variety of price ranges for
juniors
...
The demand for
Levi’s jeans is inversely proportional to the price
...
Sales
...
The Silver Tab Baggy jeans sell for about $30,
whereas the Offender jeans sell for about $160
...
If 400,000 pairs of the Silver Tab Baggy jeans were
bought, approximately how many of the Offender jeans
were bought?
150,000 km
Mars
x
SUN
228,000 km
49
...
The solar radiation on the Earth is
approximately 1400 watts per square meter (w/m2 )
...
50
...
The solar radiation on the Earth is
approximately 1400 watts per square meter
...
51
...
Marilyn receives a $25,000 bonus from her
company and decides to put the money toward a new car
that she will need in two years
...
She
compares two different banks’ rates on money market
accounts
...
What is the interest rate on
money market accounts at both banks?
52
...
Connie and Alvaro sell their house and buy a
fixer-upper house
...
They know it will take 6 months before the
general contractor will start their renovation, and they want
to take advantage of a 6-month CD that pays simple interest
...
Chemistry
...
If the
temperature decreases to 275 K, what is the resulting pressure?
54
...
A gas contained in a 4 milliliter container at a
temperature of 300 K has a pressure of 1 atmosphere
...
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■
5:00 PM
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C H A P T E R 3 Functions and Their Graphs
C AT C H T H E M I S TA K E
In Exercises 55 and 56, explain the mistake that is made
...
y varies directly with t and indirectly with x
...
Find an equation that describes this variation
...
y varies directly with t and the square of x
...
Find an equation that describes this
variation
...
y ϭ ktx
Solution:
Let x ϭ 4, t ϭ 2, and y ϭ 1
...
Solve for k
...
Let x ϭ 4, t ϭ 1, and y ϭ 8
...
1
y = tx
8
into y ϭ ktx
...
What mistake was made?
This is incorrect
...
57
...
58
...
In Exercises 59 and 60, match the variation with the graph
...
Inverse variation
60
...
b
...
The three parameters that help classify the strength of optical
turbulence are:
■
■
C 2, index of refraction structure parameter
n
k, wave number of the laser, which is inversely proportional to
the wavelength l of the laser:
k =
■
2p
L, propagation distance
The variance of the irradiance of a laser s2 is directly proportional
to C2 , k7/6, and L11/16
...
When C 2 ϭ 1
...
55 mm,
n
the variance of irradiance for a plane wave s2 is 7
...
Find
pl
the equation that describes this variation
...
When C 2 ϭ 1
...
55 mm,
n
the variance of irradiance for a spherical wave s2 is 2
...
sp
Find the equation that describes this variation
...
qxd
11/24/11
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Page 359
3
...
S
...
(Data are from Forecast Center’s Historical Economic and Market Home Page
at www
...
com/djutil
...
)
Use the calculator STAT EDIT commands to enter the table with L1 as the oil price, L2 as the utilities stock index, L3 as
number of housing units, and L4 as the 5-year maturity rate
...
S
...
99
193
...
76
1996
18
...
85
1467
5
...
17
232
...
33
1998
16
...
29
1525
5
...
47
302
...
60
2000
27
...
14
1636
6
...
58
372
...
86
2002
19
...
71
1698
4
...
94
207
...
05
2004
32
...
94
1911
3
...
84
343
...
71
2006
JANUARY
OIL PRICE,
$ PER BARREL
65
...
84
2265
4
...
An increase in oil price in dollars per barrel will drive the
U
...
Dow Jones Utilities Stock Index to soar
...
Use the calculator commands STAT , linReg (ax ϩ b),
and STATPLOT to model the data using the least squares
regression
...
b
...
S
...
Find the
variation constant and equation of variation using x as the
oil price in dollars per barrel
...
Use the equations you found in (a) and (b) to predict the
stock index when the oil price hits $72
...
Which answer is closer to the actual
stock index of 417? Round all answers to the nearest
whole number
...
An increase in oil price in dollars per barrel will affect the
interest rates across the board—in particular, the 5-year
Treasury constant maturity rate
...
Use the calculator commands STAT , linReg (ax ϩ b),
and STATPLOT to model the data using the least-squares
regression
...
b
...
Find the
variation constant and equation of variation using x as the
oil price in dollars per barrel
...
Use the equations you found in (a) and (b) to predict the
maturity rate when the oil price hits $72
...
Which answer is closer to the
actual maturity rate at 5
...
c03d
...
An increase in interest rates—in particular, the 5-year
Treasury constant maturity rate—will affect the number of
new, privately owned housing units
...
An increase in the number of new, privately owned housing
units will affect the U
...
Dow Jones Utilities Stock Index
...
Use the calculator commands STAT , linReg (ax 1 b),
and STATPLOT to model the data using the least-squares
regression
...
a
...
Find the equation of the
least-squares regression line using x as the number
of housing units
...
If the number of new privately owned housing units
varies inversely as the 5-year Treasury constant
maturity rate, then use the calculator commands STAT ,
PwrReg , and STATPLOT to model the data using the
power function
...
b
...
S
...
Find the variation constant and equation of variation
using x as the number of housing units
...
Use the equations you found in (a) and (b) to predict the
number of housing units when the maturity rate is 5
...
Which answer is closer to the actual
number of new, privately owned housing units of 1861?
Round all answers to the nearest unit
...
Use the equations you found in (a) and (b) to predict
the utilities stock index if there are 1861 new,
privately owned housing units in September 2006
...
For Exercises 67 and 68, refer to the following:
Data for retail gasoline price in dollars per gallon for the period March 2000 to March 2008 are given in the following
table
...
S
...
eia
...
gov/oog/info/gdu/gaspump
...
) Use the calculator STAT EDIT command to enter the table below
with L 1 as the year (x ϭ 1 for year 2000) and L 2 as the gasoline price in dollars per gallon
...
517
1
...
249
1
...
736
2
...
425
2
...
244
67
...
Use the calculator commands STAT LinReg to
model the data using the least-squares regression
...
Round all answers to three decimal
places
...
a
...
Find the
variation constant and equation of variation using x as the
year (x ϭ 1 for year 2000) and y as the gasoline price in
dollars per gallon
...
b
...
Round all answers to three decimal places
...
Use the equation to predict the gasoline price in March
2006
...
Is the
answer close to the actual price?
c
...
Round all answers to three decimal
places
...
Use the equation to predict the gasoline price in
March 2009
...
11/25/11
5:15 PM
Page 361
C H A P T E R 3 I N Q U I R Y- B A S E D L E A R N I N G P R O J E C T
Being a creature of habit, Dylan usually sets out
each morning at 7 AM from his house for a jog
...
a
...
Distance from home (miles)
Transformations of Functions
5
Dylan's Jog on Friday
4
3
2
y = d(t)
1
0
10 20 30 40 50 60 70 80
Time (minutes)
Figure 1
t
y = d(t)
b
...
It is a transformation
of the function y = d(t) shown in Figure 1
...
You may find it helpful to
refer to the table in part (a)
...
5
Dylan's Jog on Saturday
4
3
2
1
0
10 20 30 40 50 60 70 80
Time (minutes)
Figure 2
t
y
What is the real-world meaning of this transformation? How is Dylan’s jog on
Saturday different from his usual jog? How is it the same?
The original function (in Figure 1) is represented by the equation y = d(t)
...
Explain
...
The graph shown in Figure 3 represents
Dylan’s jog on Sunday
...
Complete the table of values below for
this transformation
...
qxd
5
Dylan's Jog on Sunday
4
3
2
1
0
10 20 30 40 50 60 70 80
Time (minutes)
Figure 3
t
y
What is the real-world meaning of this transformation? How is Dylan’s jog on Sunday
different from his usual jog?
361
c03d
...
Suppose Dylan’s jog on Monday can be
represented by the equation y = 1 d(t)
...
Distance from home (miles)
The original function (in Figure 1) is represented by the equation y = d(t)
...
Explain
...
Suppose Dylan has a goal of cutting his
usual jogging time in half, while covering
the same distance
...
Complete the table, sketch a
graph, and write an equation in function
notation
...
Finally, discuss whether you think
Dylan’s goal is realistic
...
qxd
12/27/11
2:15 PM
Page 363
MODELING OUR WORLD
The U
...
National Oceanic and Atmospheric Association (NOAA) monitors temperature and
carbon emissions at its observatory in Mauna Loa, Hawaii
...
The data presented in this chapter is from the Mauna Loa Observatory, where historical
atmospheric measurements have been recorded for the last 50 years
...
The following table summarizes average yearly temperature in degrees Fahrenheit ؇F
and carbon dioxide emissions in parts per million (ppm) for Mauna Loa, Hawaii
...
45
43
...
61
43
...
66
45
...
9
320
...
7
331
...
7
345
...
53
47
...
86
46
...
2
360
...
4
379
...
Plot the temperature data with time on the horizontal axis and temperature on the
vertical axis
...
2
...
a
...
b
...
c
...
3
...
a
...
b
...
c
...
4
...
a
...
b
...
c
...
5
...
6
...
Let t ϭ 0 correspond to 1960
...
Find a linear function that models the CO2 emissions (ppm) in Mauna Loa
...
Use data from 1965 and 1995
...
Use data from 1960 and 1990
...
Use linear regression and all data given
...
Predict the expected CO2 levels in Mauna Loa in 2020
...
Apply the line found in Exercise 7(a)
...
Apply the line found in Exercise 7(b)
...
Apply the line found in Exercise 7(c)
...
Predict the expected CO2 levels in Mauna Loa in 2100
...
Apply the line found in Exercise 7(a)
...
Apply the line found in Exercise 7(b)
...
Apply the line found in Exercise 7(c)
...
Do you think your models support the claim of the “greenhouse effect”? Explain
...
qxd
11/25/11
5:15 PM
CHAPTER 3
S ECTION
REVIEW
C ONCEPT
3
...
Function notation
Placeholder notation:
f (x) = 3x2 - 6x + 2
f(n) = 3(n)2 - 6(n) + 2
Difference quotient:
ƒ(x + h) - ƒ(x)
;h Z 0
h
Domain of a function
3
...
Are there any restrictions on x?
Graphs of functions;
piecewise-defined functions;
increasing and decreasing
functions; average rate of change
Recognizing and classifying functions
3
f (x) = x3, f (x) = 1x, f (x) = 1x,
1
ƒ(x) = ƒ x ƒ , ƒ(x) =
x
Even and odd functions
Even: Symmetry about y-axis: ƒ(Ϫx) ϭ ƒ(x)
Odd: Symmetry about origin: ƒ(Ϫx) ϭ Ϫƒ(x)
Common functions
CHAPTER REVIEW
ƒ(x) ϭ mx ϩ b, ƒ(x) ϭ x, ƒ(x) ϭ x2,
Increasing and decreasing
functions
Average rate of change
Points of discontinuity
Graphing techniques:
Transformations
Shift the graph of ƒ(x)
...
3
• Increasing: rises (left to right)
• Decreasing: falls (left to right)
cƒ(x) if c Ͼ 1; stretch vertically
cƒ(x) if 0 Ͻ c Ͻ 1; compress vertically
ƒ(cx) if c Ͼ 1; compress horizontally
ƒ(cx) if 0 Ͻ c Ͻ 1; stretch horizontally
x1 Z x2
c03d
...
4
K EY I DEAS/F ORMULAS
Operations on functions
and composition of functions
Adding, subtracting,
multiplying, and
dividing functions
Composition of functions
3
...
f
f (x)
, g(x) Z 0
a b(x) =
g
g(x)
Domain of the quotient is the intersection of
the domains of ƒ and g, and any points when
g(x) ϭ 0 must be eliminated
...
Values for x
must be eliminated if their corresponding
values g(x) are not in the domain of ƒ
...
• ƒϪ1 (ƒ(x)) ϭ x and ƒ(ƒϪ1 (x)) ϭ x
...
Range of ƒ ϭ domain of ƒϪ1
...
• If the point (a, b) lies on the graph of a
function, then the point (b, a) lies on the
graph of its inverse
...
6
• No two x-values map to the same y-value
...
• A horizontal line may intersect a one-to-one
function in at most one point
...
Let y ϭ ƒ(x)
...
Interchange x and y
...
Solve for y
...
Let y ϭ ƒϪ1(x)
...
Combined: Direct variation and inverse variation
occur at the same time
...
qxd
11/24/11
5:00 PM
CHAPTER 3
Page 366
REVIEW EXERCISES
3
...
14
...
1
...
ƒ(Ϫ5) b
...
x, where ƒ(x) ϭ 0
a
...
ƒ(4)
c
...
{(1, 2), (3, 4), (2, 4), (3, 7)}
Evaluate the given quantities using the following three
functions
...
{(Ϫ2, 3), (1, Ϫ3), (0, 4), (2, 6)}
4
...
x2 ϩ y2 ϭ 36
7
...
y = 1x
ƒ(x) ϭ 4x Ϫ 7
6
...
15
...
y
F(0)
g(0)
19
...
f(3 + h)
21
...
F(4)
17
...
y
F(t) ϭ t2 ϩ 4t Ϫ 3
f(3 + h) - f (3)
h
22
...
Express the domain in
interval notation
...
ƒ(x) ϭ Ϫ3x Ϫ 4
27
...
h(x) =
Use the graphs of the functions to find:
11
...
y
1
x + 4
24
...
H(x) =
12x - 6
26
...
29
...
Construct a function that is undefined at x ϭ Ϫ3 and x ϭ 2
such that the point (0, Ϫ4) lies on the graph of the function
...
2 Graphs of Functions
a
...
ƒ(1)
c
...
ƒ(Ϫ4) b
...
x, where ƒ(x) ϭ 0
Determine whether the function is even, odd, or neither
...
ƒ(x) ϭ 2x Ϫ 7
33
...
ƒ(x) ϭ x1/4 ϩ x
1
37
...
f (x) = 1x + 4
1
38
...
g(x) ϭ 7x5 ϩ 4x3 Ϫ 2x
34
...
qxd
11/24/11
5:00 PM
Page 367
3
51
...
Domain
b
...
Intervals on which the function is increasing, decreasing,
or constant
...
52
...
y = - 1 x3
2
54
...
y
y
x
10
y
x
–10
–10
10
x
x
–10
41
...
42
...
x2
2
y ϭ ƒ(x Ϫ 2)
57
...
y
x 6 0
x Ú 0
-2x - 3
44
...
ƒ(x) = u - 2x
ƒ x + 2ƒ
x2
46
...
y ϭ Ϫ2ƒ(x)
Applications
59
...
Tutoring Costs
...
00 for the
first hour of tutoring and $10
...
Find the cost function C(x) as a function of the length
of the tutoring session
...
y ϭ ƒ(x) ϩ 3
60
...
Salary
...
00 per hour also earns
time and a half for overtime (any hours worked above the normal
40-hour work week)
...
3
...
Shifted to the right two units and up three units
62
...
Stretched by a factor of 5 and shifted down six units
64
...
Transform the function into the form ƒ(x) ؍c(x ؊ h)2 ؉ k by
completing the square and graph the resulting function using
transformations
...
y ϭ Ϫ(x Ϫ 2)2 ϩ 4
65
...
y ϭ ƒ Ϫx ϩ 5 ƒ Ϫ 7
66
...
State the domain and
range in interval notation
...
F(x) = b
1
- 4
x - 2
56
...
10
Review Exercises
c03d
...
4 Operations on Functions and
Composition of Functions
g
Given the functions g and h, find g ؉ h, g ؊ h, g ؒ h, and ,
h
and state the domain
...
g(x) ϭ Ϫ3x Ϫ 4
h(x) ϭ x Ϫ 3
81
...
f (x) =
x - 1
3
g(x) = 1x - 4
g(x) = x2 - 1
83
...
g(x) ϭ 2x ϩ 3
h(x) = 1x
x + 3
70
...
g(x) = 2
x
2
71
...
g(x) = x2 - 4
h(x) = x + 2
For the given functions f and g, find the composite functions
f ؠg and g ؠf, and state the domains
...
ƒ(x) ϭ 3x Ϫ 4
g(x) ϭ 2x ϩ 1
4
x2 - 2
1
g (x) = 2
x - 9
84
...
f (x) = 22x2 - 5
g(x) = 1x + 6
77
...
f (x) =
1
80
...
79
...
3
1 - 2x
1
85
...
h(x) =
2x2 + 7
88
...
h(x) =
Applications
89
...
A rain drop hitting a lake makes a circular ripple
...
74
...
f (x) =
x + 3
1
g(x) =
4 - x
x
ƒ2x - 3ƒ
90
...
Let the area of a rectangle be given by
42 ϭ l ؒ w, and let the perimeter be 36 ϭ 2 ؒ l ϩ 2 ؒ w
...
3
...
91
...
qxd
11/24/11
5:00 PM
Page 369
Review Exercises
92
...
6 Modeling Functions Using
Range
Variation
Function
Write an equation that describes each variation
...
C is directly proportional to r
...
114
...
V ϭ 12h when
w ϭ 6 and l ϭ 2
...
A varies directly with the square of r
...
D
116
...
F ϭ 20p when
l ϭ 10 mm and L ϭ 10 km
...
{(2, 3), (Ϫ1, 2), (3, 3), (Ϫ3, Ϫ4), (Ϫ2, 1)}
Applications
94
...
y = 2x
95
...
{(Ϫ8, Ϫ6), (Ϫ4, 2), (0, 3), (2, Ϫ8), (7, 4)}
98
...
ƒ(x) ϭ x3
100
...
Wages
...
Find an equation that shows their wages (W) varying
directly with the number of hours (H) worked
...
ƒ(x) =
; ƒ -1(x) =
4x - 7
4x
101
...
ƒ(x) = 2x + 4; ƒ - 1(x) = x2 - 4
104
...
Find its inverse and check your
answer
...
107
...
f (x) =
x + 3
3
110
...
ƒ(x) ϭ x5 ϩ 2
108
...
50
Dickson
30
$255
...
Sales Tax
...
A new resident knows the difference but
doesn’t know which county has the higher tax rate
...
If the tax on a $50
...
50 in
County A and the tax on a $20
...
60 in
County B, write two equations (one for each county) that
describe the tax (T ), which is directly proportional to the
purchase price (P)
...
ƒ(x) ϭ 2x ϩ 1
HOURS WORKED
Cole
Technology Exercises
Section 3
...
Salary
...
Write a
function S(x) that represents her yearly salary as a function
of the total dollars worth of products sold x
...
What does this inverse function tell you?
112
...
Express the volume V of a rectangular box that has
a square base of length s and is 3 feet high as a function of
the square length
...
If a certain volume is desired,
what does the inverse tell you?
2x - 2x - 3
119
...
Express the domain in interval notation
...
Use a graphing utility to graph the function and find the
domain
...
f (x) =
x2 - 4x - 5
x2 - 9
REVIEW EXERCISES
Verify that the function ƒ ؊1(x) is the inverse of ƒ (x) by
showing that ƒ(ƒ ؊1(x)) ؍x
...
c03d
...
2
Section 3
...
Use a graphing utility to graph the function
...
From December 1999 to December 2007, data for gold price in
dollars per ounce are given in the table below
...
finfacts
...
htm
...
f (x) =
1 - x
[[x]]
Lx + 1
x 6 -1
-1 … x 6 2
x 7 2
122
...
State the
(a) domain, (b) range, and (c) x intervals where the function
is increasing, decreasing, and constant
...
3
REVIEW EXERCISES
123
...
Use transforms to describe the relationship
between f (x) and g (x)?
124
...
Use transforms to describe the
relationship between f (x) and g (x)?
125
...
What is the domain of y3?
y2
Section 3
...
Using a graphing utility, plot y1 = 2x2 - 4, y2 = x2 - 5,
and y3 = y2 - 5
...
The graph of y3 is only defined for the
domain of g ؠf
...
Section 3
...
Use a graphing utility to graph the functions f and g and the
line y = x in the same screen
...
Use a graphing utility to graph the function and determine
whether it is one-to-one
...
90
2000
272
...
70
2002
ƒ x2 - 1ƒ
GOLD PRICE IN
$ PER OUNCE
2001
f (x) = e
DECEMBER OF
EACH YEAR
346
...
80
2004
438
...
20
2006
636
...
20
129
...
Use the calculator commands STAT LinReg to model
the data using the least-squares regression
...
Round all answers to two decimal
places
...
Use the equation to predict the gold price in December
2005
...
Is the
answer close to the actual price?
c
...
Round all answers to two decimal places
...
a
...
Find the variation
constant and equation of variation using x as the year
(x ϭ 1 for year 1999) and y as the gold price in
dollars per ounce
...
b
...
Round all answers to two decimal places
...
Use the equation to predict the gold price in December
2008
...
c03d
...
y
3
...
not a function
b
...
a one-to-one function
y ϭ p(x)
Use f(x) 1 ؍x ؊ 2 and g(x) ؍x2 ؉ 11, and determine the
desired quantity or expression
...
1
...
x ϭ y2 ϩ 2
4
...
a b (x)
g
g
6
...
g( f (x))
9
...
(ƒ ϩ g)(6)
Determine whether the function is odd, even, or neither
...
ƒ(x) ϭ ƒ x ƒ Ϫ x
11
...
f(x) = - 1x - 3 + 2
2
12
...
State the domain and range of each
function
...
f (x) = 3x2 - 4x + 1
20
...
f (x) = 1x - 1 for x = 2 to x = 10
Find the average rate of change of the given functions
...
f (x) = 64 - 16x2 for x = 0 to x = 2
23
...
State the
domain and range of both ƒ and ƒϪ1
...
Find
25
...
f(x) ϭ x2 ϩ 5
26
...
What domain restriction can be made so that ƒ(x) ϭ x2 has an
inverse?
y ϭ f(x)
x
28
...
Discount
...
An advertisement in the newspaper has an
“additional 30% off the sale price” coupon
...
a
...
ƒ(0)
d
...
c
...
x, where ƒ(x) ϭ 0
y
31
...
If a quarter circle is drawn by tracing the unit circle
in quadrant III, what does the inverse of that function look
like? Where is it located?
y ϭ g(x)
x
a
...
g(0)
d
...
Temperature
...
15
...
c
...
Sprinkler
...
The puddle of water forms a circular
pattern around the sprinkler head with a radius in yards that
grows as a function of time, in hours: r (t) = 101t
...
)
371
P R ACTI C E TE ST
-x
15
...
ƒ(x) ϭ Ϫ2(x Ϫ 1)2
b
...
p(3)
a
...
p(1)
c03d
...
Internet
...
Write a function describing the cost of the service as a
function of minutes used
...
34
...
y ϭ 8 when x ϭ 5
...
F varies directly with m and inversely with p
...
36
...
State the
(a) domain, (b) range, and (c) x intervals where the function
is increasing, decreasing, and constant
...
Use a graphing utility to graph the function and determine
whether it is one-to-one
...
qxd
11/24/11
5:00 PM
3 - 25
Page 373
CHAPTERS 1—3
1
...
Write an equation of a line that passes through the points
(1
...
2, Ϫ3)
...
2
...
3
...
Transform the equation into standard form by completing the
square, and state the center and radius of the circle:
x2 ϩ y2 ϩ 12x Ϫ 18y Ϫ 4 ϭ 0
...
x + 2
19
...
1
1
4
...
6
5
5
...
6
...
- 10 =
x
3x
7
...
50
...
70
...
8
...
1
x2
- x =
...
Solve using substitution: x Ϫ x Ϫ 12 ϭ 0
...
12
...
x
6 0
x - 5
14
...
7 Ϫ 3
...
3
15
...
7, Ϫ1
...
2, 6
...
21
...
g(x) =
x - 1
22
...
23
...
24
...
25
...
r ϭ 45 when t ϭ 3
...
Use a graphing utility to graph the function
...
f (x) = e
1 - ƒxƒ
1 - ƒ x - 2ƒ
-1 … x 6 1
1 6 x…3
27
...
Find the function
h such that g ؠh = f
...
Find the slope of the line passing through the points
(0
...
4) and (2
...
3)
...
Solve and check: 2x + 2 = -3
...
Solve by completing the square:
20
...