164
(3-44)
Chapter 3
Graphs and Functions in the Cartesian Coordinate System
a) Write an inequality to describe this region.
b) Will the truck hold 71 refrigerators and 118 TVs?
c) Will the truck hold 51 refrigerators and 176 TVs?
a) 3r t 330 b) no c) yes
400
Number of TVs
(0, 330)
300
200
GET TING MORE INVOLVED
100
75. Writing. Explain the difference between a compound inequality using the word and and a compound inequality
using the word or.
76. Discussion. Explain how to write an absolute value
inequality as a compound inequality.
(110, 0)
0
0
50
100
150
Number of refrigerators
FIGURE FOR EXERCISE 74
3.5
In this
section
●
Definitions
●
Domain and Range
●
The Rule Definition of
Function
●
Function Notation or
f-notation
●
Average Rate of Change
RELATIONS AND FUNCTIONS
Earlier in this chapter we used the phrase “is a function of ” to describe a special relationship between variables. The area of a circle is a function of its radius because
the area is determined by the radius using A r2. The area of rectangle is not a
function of its length because the area is not determined by the length alone. So “is
a function of” means “is determined by.” In this section we will learn that function
is also a noun. A function is a special kind of set. By studying functions as sets, we
can make the concept of functions more precise.
Definitions
Apple Imagewriter ribbons are sold in boxes of six in the K-LOG Catalog and are
priced as follows.
Number of boxes
Cost per ribbon
1
2–3
4
$4.85
$4.60
$4.35
We can write this data as a set of ordered pairs in which the first coordinate is the
number of boxes purchased and the second is the cost per ribbon in dollars:
(1, 4.85), (2, 4.60), (3, 4.60), (4, 4.35)
helpful
hint
The key word here is “determines.” According to the dictionary, determine means to
settle conclusively. If the second coordinate of an ordered
pair is inconclusive, then the
set of ordered pairs is not a
function.
Because the number of boxes determines the cost per ribbon, we say that the cost is
a function of the number of boxes purchased.
Suppose the following table appeared in the K-LOG Catalog:
Number of boxes
Cost per ribbon
1
2
2
4
$4.85
$4.60
$4.45
$4.35
Something is wrong with this table. The cost per ribbon when you buy two boxes is
not clear because the ordered pairs (2, 4.60) and (2, 4.45) have the same first coordinate and different second coordinates. In this case the cost per ribbon is not a
function of the number of boxes.
These examples illustrate the definition of function.
Function—A Set of Ordered Pairs
A function is a set of ordered pairs in which no two ordered pairs have the
same first coordinate and different second coordinates.
3.5
Relations and Functions
(3-45)
165
Any set of ordered pairs is called a relation. So every function is a relation, but
a relation in which two ordered pairs have the same first coordinate and different
second coordinates is not a function.
E X A M P L E
helpful
1
hint
For an example of a function
in real life consider the function that pairs up each universal product code at your
grocery store with the price of
the item.
E X A M P L E
helpful
2
hint
Some people like to think of a
function as a machine. The
first coordinate (x) is put into
the machine, the handle is
turned, and the second coordinate (y) comes out.
x
Functions defined from a list of ordered pairs
Determine whether each relation is a function.
a) (1, 2), (1, 5), (3, 7)
b) (4, 5), (3, 5), (2, 6), (1, 7)
Solution
a) This relation is not a function because (1, 2) and (1, 5) have the same first coordinates but different second coordinates.
b) This relation is a function. Note that the same second coordinate with different
■
first coordinates is permitted in a function.
If variables are used to represent the numbers of the ordered pairs, then the variable for the first coordinate is the independent variable and the variable for the
second coordinate is the dependent variable. When we use variables x and y, we
always assume x is the independent variable and y is the dependent variable. If we
state that B is a function of A, then A is the independent variable (first coordinate)
and B is the dependent variable (second coordinate), and we write (A, B).
Functions defined with set notation
Determine whether each relation is a function.
a) (x, y) y 5x2 7x 2 b) (s, t) t2 s
c) (u, v) 2u 3v 6
Solution
a) This relation is a function, or y is a function of x, because for any x there is only
one value for y determined by the equation y 5x2 7x 2.
b) Because the dependent variable is squared in t2 s, a positive value for s corresponds to both a positive and a negative value for t. For example, if s 4, then
t 2 4, or t 2. The ordered pairs (4, –2) and (4, 2) belong to this set. So this
relation is not a function; that is, t is not a function of s.
2
c) If we solve 2u 3v 6 for v, we get v 3 u 2. Because each value
of u determines only one value for v, this relation is a function. In this case, v is
■
a linear function of u.
We often omit the set notation and refer to an equation as a relation or a function. There are many well-known formulas that express the value of one variable as
a function of another variable. For example, A r 2 gives the area of a circle
as a function of its radius. The formula F 9C 32 expresses the Fahrenheit
5
temperature as a linear function of the Celsius temperature.
y
Any equation with two variables is a relation because it deterCAUTION
mines a set of ordered pairs. However, an equation is a function only if the set of
ordered pairs that satisfy the equation is a function.
E X A M P L E
3
Functions defined by equations
Determine whether each relation defines y as a function of x.
b) y x 2
c) x 2 y 2 16
a) x y4
166
(3-46)
helpful
Chapter 3
hint
In a function every value for
the independent variable
determines conclusively a
corresponding value for the
dependent variable. If there is
more than one possible value
for the dependent variable,
then the set of ordered pairs is
not a function.
Graphs and Functions in the Cartesian Coordinate System
Solution
a) Because the dependent variable appears to the fourth power in x y 4, a positive
value for x corresponds to both a positive and a negative value for y. For example, both (1, 1) and (1, 1) satisfy x y 4, and these ordered pairs have the same
first coordinate and different second coordinates. So x y4 does not define y as
a function of x. The equation x y4 is not a function.
b) For each value of x the equation y x2 determines only one value for y. So y x 2 defines y as a function of x. We also say that y x 2 is a function. Note that
(1, 1) and (1, 1) satisfy y x 2, but ordered pairs with the same second coordinate and different first coordinates are allowed in a function.
c) The even power of the dependent variable in x 2 y 2 16 indicates that we can
use a number or its opposite for y and get the same value for y 2. For example, if
y 4, then both (0, 4) and (0, 4) satisfy x 2 y2 16. So the equation is not
■
a function. The equation does not define y as a function of x.
Domain and Range
The domain of a relation (or function) is the set of first coordinates, and the range
is the set of second coordinates of the ordered pairs. If the ordered pairs of a relation
are listed, then the domain and range can be read from the list. Relations are often
defined by equations with no domain stated. If the domain is not stated, we agree
that the domain consists of all real numbers that, when substituted for the independent variable, produce real numbers for the dependent variable.
E X A M P L E
4
Identifying domain and range
State the domain and range of each relation.
a) (2, 5), (2, 7), (4, 3)
b) y x
Solution
a) The domain is the set of numbers used as first coordinates, 2, 4. The range is the
set of second coordinates, 3, 5, 7.
b) Because x is a real number only for x 0, the domain is [0, ), the nonnegative real numbers. The range is the set of numbers that result from taking the principal square root of every nonnegative real number. Thus the range is also the
■
set of nonnegative real numbers, [0, ).
For some relations it is easier to determine the domain than the range. In Section 3.6 we will see that the graph of a relation can be helpful for finding the range.
The Rule Definition of Function
There is another definition of function that is equivalent to the ordered-pair definition of function that we have been using:
Function—A Rule
A function is a rule that assigns to each element of one set (the domain)
exactly one element of another set (the range).
If a set of ordered pairs is a function, then the ordered pairs give us a rule for assigning each element of one set with exactly one element of another set. Conversely,
3.5
Relations and Functions
(3-47)
167
if we have a rule for assigning elements, then we could write all of the assignments
as ordered pairs that satisfy the ordered-pair definition of function. So a function by
one definition is also a function by the other definition.
Function Notation or f-notation
If concert tickets are $30 each, we write the cost of n tickets as C 30n. We could
also write C(n) 30n, where C(n) is read as “C of n.” If n 5, then C(5) $150.
A third way to express this function is to write C f (n), which is read “C equals f of
n or C is a function of n.” With this notation, f is not thought of as a variable, but
rather as a name for the function that pairs a value for n with a value for C. This notation is called function notation or f-notation. If we are discussing two functions,
we can name them with different letters. For example, if f (x) x 2 x 9
and g(x) 3x 1, we can refer to the functions as f and g without mentioning the
formulas. To find f (4), we write
f (4) 42 4 9 11.
The notation f(4) does not mean f times 4.
CAUTION
E X A M P L E
5
calculator
Using f-notation
Let f(x) 3x 2 and g(x) x 2 x. Find the following:
a) f (5)
b) g(3)
c) x, if f (x) 0
Solution
a) Replace x by 5 in the equation defining the function f :
close-up
A graphing calculator has a
built-in f-notation. Define y1 3x 2 and y2 x 2 x using
Y .
So f (5) 17.
f (x) 3x 2
f (5) 3(5) 2
17
b) Replace x by 3 in the equation defining the function g:
So g(3) 12.
g(x) x2 x
g(3) (3)2 (3)
12
c) Because f (x) 3x 2 and we are given that f (x) 0, we can conclude that
3x 2 0. To find x, solve this equation:
Now use the variables feature
(VARS) to find y1(5) and
y2(3).
3x 2 0
3x 2
2
x
3
■
The letter used for the independent variable in f (x) x 2 2 is not important.
The notation f (t) t 2 2 or even
f (first coordinate) (first coordinate)2 2
could be used to define the same set of ordered pairs. The second coordinate of an
ordered pair of this function is determined by squaring the first coordinate and
adding 2.
168
(3-48)
Chapter 3
Graphs and Functions in the Cartesian Coordinate System
The first coordinate of an ordered pair can be an expression rather than a real
number. For example, if the first coordinate is the expression a 4 and the function is f(x) 3x 2, then
f (a 4) 3(a 4) 2
3a 12 2
3a 14
Replace x by a 4.
Distributive property
The first coordinate can even be an expression involving x. For example, if the first
coordinate is x 5 and the function is f (x) 3x 2, then
f(x 5) 3(x 5) 2
3x 17
Replace x by x 5.
Note that x 5 replaces x and x 5 is multiplied by 3, not just x.
E X A M P L E
6
Using an expression for the independent variable
Let f(x) 3x 2 and g(x) 1x. Find the following:
a) f (a 3)
b) g(x h)
Solution
a) Replace x by a 3 in the equation defining the function f :
f (x) 3x 2
f (a 3) 3(a 3) 2
3a 11
So f(a 3) 3a 11.
b) Replace x by x h in the equation defining the function g:
1
x
g(x) f(x) = 2 x – 7
f
5
3
x
f (x)
Domain of f
Range of f
FIGURE 3.34
1
xh
g(x h) ■
A function is a rule for pairing numbers in one set (the domain) with numbers in
another set (the range). Figure 3.34 shows the domain and range of a function. The
arrow represents the function and illustrates the pairing of values in the domain with
values in the range.
Average Rate of Change (Optional)
The U.S. Bureau of Labor Statistics reports on the number of people (in millions) in
the civilian labor force of the United States. The size of the labor force is a function
of the year. Two ordered pairs of this function are (1985, 88.4) and (1989, 97.3). To
see how the function is changing in time, we can find the slope of the line through
these two points:
97.3 88.4
2.2 million people per year
1989 1985
During this 4-year period the civilian labor force increased on the average by
2.2 million people per year. Note that 2.2 million is not necessarily the increase for
any particular year, but it is the average rate at which the labor force was changing
during those 4 years.
3.5
y
Relations and Functions
(3-49)
169
In general, the average rate of change of a function over an interval in its domain
is defined as follows.
f
f(b)
Average Rate of Change of a Function
f (b) – f (a)
If a and b are in the domain of the function f and a b, then the average
rate of change of f over the interval [a, b] is
f(a)
b–a
a
f (b) f(a)
.
ba
x
b
FIGURE 3.35
Note that the average rate of change of f over [a, b] is the slope of the line through
(a, f (a)) and (b, f(b)), as shown in Fig. 3.35.
E X A M P L E
7
Average rate of change of a function (Optional)
If a stunt man jumps from a bridge that is 144 feet above the water, then his height
above the water in feet at time t in seconds is given by the function
h(t) 16t2 144. Find the average rate of change of his height above the water
over the time interval [1, 3]. See Fig. 3.36.
h(t)
150
120
90
60
30
0
(1, 128)
Solution
Because h(1) 128 and h(3) 0, we have
h(3) h(1) 0 128
31
2
(3, 0)
1
2
t
3
64 ft/sec.
FIGURE 3.36
So the average rate of change of the height is 64 ft /sec on the interval [1, 3]. ■
WARM-UPS
True or false? Explain your answer.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
3. 5
Any set of ordered pairs is a function. False
The circumference of a circle is a function of the diameter. True
The set (1, 2), (3, 2), (5, 2) is a function. True
Every relation is a function. False
The set (1, 5), (3, 6), (1, 7) is a function. False
The domain of f (x) x is the set of positive real numbers. False
The range of g(x) x is the set of nonnegative real numbers. True
The set (x, y) x 4y is a function. True
The set (u, v) u v2 is a function. False
If h(x) x2 3, then h(2) 1. True
EXERCISES
Reading and Writing After reading this section, write out the
answers to these questions. Use complete sentences.
1. What is a function?
A function is a set of ordered pairs in which no two have the
same first coordinate and different second coordinates.
2. What is a relation?
A relation is any set of ordered pairs.
3. What is the domain of a relation?
The domain of a relation is the set of all possible first
coordinates.
170
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Chapter 3
Graphs and Functions in the Cartesian Coordinate System
4. What is the range of a relation?
The range of relation is the set of all possible second
coordinates.
5. What is f-notation?
The f-notation is the notation in which we use f(x) rather
than y as the dependent variable.
6. What is the average rate of change of a function?
The average rate of change of the function f(x) on the interval [a, b] is ( f(b) f(a))
(b a).
Determine whether each relation is a function. See Example 1.
7. (2, 4), (3, 4), (4, 5)
8. (2, 5), (2, 5), (3, 10)
Yes
No
9. (2, 4), (2, 6), (3, 6)
10. (3, 6), (6, 3)
No
Yes
11. (, 1), (, 1) No
12. (0.3, 0.3), (0.2, 0), (0.3, 1) No
1 1
13. , Yes
2 2
1
1
1
Yes
14. , 7 , , 7 , , 7
3
3
6
Determine whether each relation is a function. See Example 2.
15. (x, y) y (x 1)2 Yes
16. (x, y) y x2 12x 1 Yes
17. (x, y) x y No
18. (x, y) x y2 2 No
1
19. (s, t) t s Yes
20. (u, v) v Yes
u
21. (x, y) x 5y 2 Yes 22. (x, y) x 3y Yes
Determine whether each relation defines y as a function of x.
See Example 3.
23. x 2y2
24. y 3x4
25. y 3x 4
No
Yes
Yes
26. y 2x 9
27. x2 y2 1
28. x 4 y 4 0
Yes
No
No
29. y x
30. x y
31. x 2 y Yes
Yes
No
32. y x 1 Yes
Determine the domain and range of each relation. See
Example 4.
33. (2, 3), (2, 5), (2, 7)
34. (3, 1), (5, 1), (4, 1)
2, 3, 5, 7
3, 4, 5, 1
35. y x
36. y 2x 1
R, [0, )
R, R
37. x y
38. y x 1
[0, ), [0, )
[0, ), [1, )
Let f(x) 3x 2, g(x) x2 3x 2, and h(x) x 2 .
Find the following. See Example 5.
39. f(4) 10
40. f(100) 298
41. g (2) 12
42. g(6) 20
43. h(3) 1
44. h(19) 17
45. x, if f(x) 5
7
3
46. x, if f(x) 49
17
47. x, if h(x) 3
5 or 1
48. x, if h(x) 7
5 or 9
1
Let f(x) 4x 1 and g(x) x
. Find the following. See
2
Example 6.
49. f(a)
50. f(a 1)
51. f(x 2)
4a 1
4a 3
4x 7
52. f(x h)
53. g(x 3)
54. g(x 2)
1
1
4x 4h 1
x
x5
55. g(x h)
56. g(a 2)
1
1
xh2
a
Find the average rate of change of the given function over the
given interval. See Example 7.
58. g(x) 2x 5, [3, 9] 2
57. f(x) x2, [1, 3] 4
59. h(x) 2x 2 3x, [4, 8] 21 60. f(x) x, [4, 9] 1
5
1
2
61. g(x) , [2, 4] 1
62. h(x) ,
[1, 2] 3
2
8
2
x
x
2
Let f(x) x
2 and g(x) 3x 8x 2. Use a
calculator to find the following. Round answers to
three decimal places.
63. f(3.46) 2.337
64. g(1.37) 18.591
65. g(3.5) 66.75
66. f(1.2) 0.894
Let f(x) 3x 2 and g(x) 3 5x. Find and simplify each
expression.
67. f(a 5) f(a) 15
68. f(x h) f (x) 3h
69. g(a 2) g(a) 10
70. g(x h) g(x) 5h
f(x h) f (x)
f(n 3) f(n)
71. 3
72. 3
h
3
Solve each problem.
73. Area of a square. Express the area of a square, A, as a
function of its side, s. A s2
74. Perimeter of a square. Express the perimeter of a square,
P, as a function of its side, s. P 4s
75. Cost of fabric. If a certain fabric is priced at $3.98 per
yard, express the cost of a purchase, C, as a function of the
number of yards purchased, y. C 3.98y
76. Earned income. If Mildred earns $14.50 per hour, express
her total pay, P, as a function of the number of hours
worked, h. P 14.5h
77. Cost of pizza. A pizza parlor in Victoria, B.C. charges
$14.95 for a pizza plus $0.50 for each topping. Express the
cost of a pizza, C, as a function of the number of
toppings, n. C 0.50n 14.95
78. Dealing in gravel. A gravel dealer charges $120 for a minimum load of 9 cubic yards and $10 more for each additional
cubic yard. Express the total charge, C, as a function of the
number of yards sold, n, where n 9. C 10n 30
79. Staying fit. Suppose the heart rate of a certain individual is
a linear function of the number of minutes she spends on a
treadmill. A heart rate of 78 was measured after 2 minutes,
and a heart rate of 86 was measured after 4 minutes. Write
the heart rate, h, as a linear function of the number of minutes, t, on the treadmill, for 0 t 8. h 4t 70
3.6
81. Depreciating Beretta. A Chevrolet Beretta that sold new
for $11,640 in 1990 sold used for $3,590 in 1998
(Edmund’s Used Car Prices, www.edmunds.com). Find the
average rate of change of the value of this car over the time
interval [1990, 1998].
$1,006.25 per year
82. More depreciation. A Porsche 928S that sold new for
$69,680 in 1988 sold used for $16,550 in 1998 (Edmund’s
Used Car Prices, www.edmunds.com). Find the average
rate of change of the value of this car over the time interval
[1988, 1998].
$5,313 per year
Speed (miles per hour)
83. Fast cat. The 1999 Mercury Cougar had a base price of
$17,095 and could go from 0 to 60 mph in 8.0 seconds
(Fortune, June 26, 1998, www.pathfinder.com). Find the
average rate of change of the velocity of this car over the
time interval [0, 8.0].
7.5 mph per second
60
40
20
0
0
2
4
6
Time (seconds)
8
FIGURE FOR EXERCISE 83
3.6
In this
section
●
Linear and Constant
Functions
●
Absolute Value Functions
●
Quadratic Functions
●
Square-Root Functions
●
Graphs of Relations
●
Vertical-Line Test
●
Applications
(3-51)
171
84. Total construction. For May 1996 the rate for total value
of construction put in place was $580 billion per year. For
May 1998 the rate was $634 billion per year (U.S. Census
Bureau, www.census.gov). Find the average rate of change
of the rate for this 24-month period, [0, 24].
$2.25 billion per year per month
Total Construction
Put in Place
680
Annual rate
(in billions of dollars)
80. Printing costs. To determine the cost of printing a book, a
printer uses a linear function of the number of pages. If the
cost is $8.60 for a 400-page book and $12.20 for a 580 page
book, then what is the linear function that is used?
C 0.02p 0.60
Graphs of Functions
640
$634
600
560
$580
May 96
May 97
May 98
FIGURE FOR EXERCISE 84
GET TING MORE INVOLVED
85. Exploration. In Example 7 we found that the stunt man
traveled 128 feet in 2 seconds for an average velocity of
64 ft/sec over the interval [1, 3]. His velocity actually
starts out at 0 ft/sec and keeps increasing as he falls. Find
the average rate of change of his height, or his average
velocity, over the time intervals [2.8, 3], [2.9, 3], and
[2.99, 3].
92.8 ft/sec, 94.4 ft/sec, 95.84 ft/sec
86. Exploration. In Exercise 85 we found the stunt man’s average velocity for some time intervals right before he hits
the water. His velocity at the moment he hits the water is his
instantaneous velocity at t 3. Find the average velocity
over the time intervals [2.999, 3] and [2.9999, 3]. What do
you think his instantaneous velocity is at time t 3?
Explain your answer.
95.984 ft/sec, 95.9984, 96 ft/sec
GRAPHS OF FUNCTIONS
Earlier in this chapter we used graphs to visualize the solution sets to linear equations. In this section we will use graphs to visualize various types of functions.
Linear and Constant Functions
Linear Function
A linear function is a function of the form
f (x) mx b,
where m and b are real numbers with m 0.