Hungerford T.W., Shaw D.J. Contemporary Precalculus: A Graphing Approach
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If a number is substituted for x in this equation, then exactly one value of y is
produced. In other words, for every real number x there exists exactly one y such
that the equation 4x 2y
3
5 0 is true.
So we can define a function whose domain is the set of all real numbers and
whose rule is
The input x produces the output
3
2x 5
/2
.
In this situation, we say that the equation defines y as a function of x.
The original equation can also be solved for x:
4x 2y
3
5
x
2y
3
4
5
.
Now if a number is substituted for y, exactly one value of x is produced. So we can
think of y as the input and the corresponding x as the output and say that the equa-
tion defines x as a function of y. ■
EXAMPLE 9
Does the equation x
2
y 1 0 define y as a function of x, or x as a function of
y, or both?
SOLUTION Solving for y, we have
x
2
1 y
y x
2
1.
This equation defines y as a function of x, since each value of x produces exactly
one value of y.
Solving for x, we obtain
x
2
y 1
x
y 1
.
This equation does not define x as a function of y because, for example, the input
y 5 produces two outputs: x 2. ■
EXAMPLE 10
A group of students drives from Cleveland to Seattle, a distance of 2350 miles, at
an average speed of 52 mph.
(a) Express their distance from Cleveland as a function of time.
(b) Express their distance from Seattle as a function of time.
SOLUTION
(a) Let t denote the time traveled in hours after leaving Cleveland, and let D be
the distance from Cleveland at time t. Then the equation that expresses D as
a function of t is
D Distance traveled in t hours at 52 mph 52t.
146 CHAPTER 3 Functions and Graphs
TECHNOLOGY TIP
To find the table setup screen, look for
TBLSET (or RANG or NUM SETUP) on
the keyboard or in the TABLE menu.
The increment is called TBL on
TI (and PITCH or NUMSTEP on others).
The table type is called INDPNT
on TI, and NUMTYPE on HP-39gs.
(b) At time t, the car has traveled 52t miles of the 2350-mile journey, so the dis-
tance K remaining to Seattle is given by K 2350 52t. This equation
expresses K as a function of t. ■
Graphing calculators are designed to deal with equations that define y
as a function of x. Calculators can evaluate such functions (that is, produce the
outputs from various inputs). One method is illustrated in the next example.
EXAMPLE 11
The equation y x
3
2x 3 defines y as a function of x. Use the table feature
of a calculator to find the outputs for each of the following inputs:
(a) 4, 3, 2, 1, 0, 1, 2, 3, 4 (b) 5, 11, 8, 7.2, .44
SOLUTION
(a) To use the table feature, we first enter y x
3
2x 3 in the equation mem-
ory, say, as y
1
. Then we call up the setup screen (see the Technology Tip in the
margin and Figure 3–4) and enter the starting number (4), the increment
(the amount the input changes for each subsequent entry, which is 1 here),
and the table type (AUTO, which means the calculator will compute all the
outputs at once).* Then press TABLE to obtain the table in Figure 3–5. To
find values that don’t appear on the screen in Figure 3–5, use the up and down
arrow keys to scroll through the table.
(b) With an apparently random list of inputs, as here, we change the table type to
ASK (or USER or BUILD YOUR OWN).
†
Then key in each value of x, and
hit ENTER. This produces the table one line at a time, as in Figure 3–6. ■
Figure 3–4 Figure 3–5 Figure 3–6
SECTION 3.1 Functions 147
EXAMPLE 12
The revenues y of MTV in year x can be approximated by the equation
y .257x
3
11.6x
2
4.5x 177.5 (0 x 14),
*There is no table type selection on Casio, but you must enter a maximum value for x.
†
Casio users should see Exercise 51.
Construct a table of values for the function in Example 11 that shows the outputs for
these inputs: 2, 2.4, 2.8, 3.2, 3.6, and 4. What is the increment here?
CALCULATOR EXPLORATION
where x 0 corresponds to 1987 and y is in millions of dollars.* So the revenue
y is a function of the year x. In what year did revenues first exceed one billion
dollars?
SOLUTION Since y is in millions, one billion dollars corresponds to y 1000.
Make a table of values for the function (Figure 3–7). It shows that the revenue was
approximately $865,550,000 in 1994 (x 7) and $1,087,500,000 in 1995
(x 8). ■
148 CHAPTER 3 Functions and Graphs
Figure 3–7
*Based on data from MTV Networks.
EXERCISES 3.1
In Exercises 1–4, determine whether or not the given table
could possibly be a table of values of a function. Give reasons
for your answer.
1.
2.
3.
4.
Exercises 5–10 deal with the greatest integer function of
Example 7, which is given by the equation y x. Compute
the following values of the function:
5. 6.75 6. .75
7. 4/3 8. 5/3
9. 16.0001
10. Does the equation y x define x as a function of y? Give
reasons for your answer.
In Exercises 11–18, determine whether the equation defines y
as a function of x or defines x as a function of y.
11. y 3x
2
12 12. y 2x
4
3x
2
2
13. y
2
4x 1 14. 5x 4y
4
64 0
15. 3x 2y 12 16. y 4x
3
14 0
17. x
2
y
2
9 18. x
2
2xy y
2
0
In Exercises 19–22, each equation defines y as a function of x.
Create a table that shows the values of the function for the
given values of x.
19. y x
2
x 4; x 2, 1.5, 1, . . . , 3, 3.5, 4.
20. y
4 x
2
; x 2, 1.2, .4, .4, 1.2, 2
21. y x
2
5; x 8, 6, . . . , 8, 10, 12
22. y x
10
100xx1, 0, 1, 2, 3, 4
23. Consider the Louisiana Tax law in Example 1. Find the out-
put (tax amount) that is produced by each of the following
inputs (incomes):
$400 $1509 $25000
$20,000 $12,500 $55,342
24. One proposed tax code looks like this:
Input 103 15
Output 232.5 2 14
Input 53 035
Output 030 53
Input 51 357
Output 024 68
Input 1 12 23
Output 1 2 5 68
Find four different numbers in the domain of this function
that produce the same output (number in the range).
25. Explain why your answer in Exercise 24 does not contradict
the definition of a function (in the box on page 144).
26. Is it possible to do Exercise 24 if all four numbers in the do-
main are required to be greater than 12,500? Why or why not?
27. The amount of postage required to mail a first-class letter is
determined by its weight. In this situation, is weight a func-
tion of postage? Or vice versa? Or both?
Income
At But Less
Least Than Tax
0 $12,500 0
$12,500 $100,000 5% of amount over $12,500
$100,000 $5,000 10% of amount
over $100,000
28. Chinese philosopher Laotze (600 BC) said, “the farther one
travels, the less one knows.” Let x be the distance one trav-
els, and y be the amount one knows. If Laotze is right, is y a
function of x? Is x a function of y? Why or Why not?
29. Could the following statement ever be the rule of a function?
For input x, the output is the
number whose square is x.
Why or why not? If there is a function with this rule, what
is its domain and range?
30. (a) Use the following chart to make two tables of values
(one for an average man and one for an average
woman) in which the inputs are the number of drinks
per hour and the outputs are the corresponding blood
alcohol contents.*
(b) Does each of these tables define a function? If so, what
are the domain and range of each function? [Remember
that you can have part of a drink.]
31. The prime rate is the rate that large banks charge their best
corporate customers for loans. The graph shows how the
prime rate charged by a particular bank has varied in recent
years.
†
Answer the following questions by reading the
graph as best you can.
(a) What was prime rate in January, 2000? In January
2001? In mid-2005?
(b) In what time period was the prime rate below 5%?
(c) On the basis of the data provided by this graph, can the
prime rate be considered a function of time? Can time
be considered a function of the prime rate?
1992
1994 1996 1998 2000 2002
Year
Prime rate (%)
x
y
2
0
4
6
8
10
2004 2006
Blood alcohol content
A look at the number of drinks consumed and blood alcohol content
in one hour under optimum conditions:
Average man
(170 lb)
Average woman
(137 lb)
.01
1 hour
.02 .03 .04 .05 .06 .07 .08 .09 .10
SECTION 3.1 Functions 149
*National Highway Traffic SafetyAdministration. Art by AP/Amy Kranz.
†
Federal Reserve Board.
32.
Find an equation that expresses the area A of a circle as a
function of its
(a) radius r (b) diameter d
33. Find an equation that expresses the area of a square as a
function of its
(a) side x (b) diagonal d
34. A box with a square base of side x is four times higher than
it is wide. Express the volume V of the box as a function
of x.
35. The surface area of a cylindrical can of radius r and height
h is 2pr
2
2prh. If the can is twice as high as the diameter
of its top, express its surface area S as a function of r.
36. Suppose you drop a rock from the top of a 400-foot-high
building. Express the distance D from the rock to the
ground as a function of time t. What is the range of this
function? [Hint: See Example 3.]
37. A bicycle factory has weekly fixed costs of $26,000. In
addition, the material and labor costs for each bicycle are
$125. Express the total weekly cost C as a function of the
number x of bicycles that are made.
38. The table below shows the percentage of single-parent fam-
ilies in various years.*
(a) The equation
y (2.4542 10
6
)x
5
(2.2459 10
4
)x
4
(.0065232)x
3
(.055795)x
2
.14568x 12.8
in which x 0 corresponds to 1960, defines y as a func-
tion of x. Make a table of values that includes the
x-values corresponding to the years in the Census
Bureau table.
(b) How do the values of y in your table compare with the
percentages in the Census Bureau table? Does this
equation seem to provide a reasonable model of the
Census Bureau data?
(c) Use the equation to estimate the percentage of single-
parent families in 1995 and 2005.
(d) Assuming that this model remains reasonably accurate,
in what year will 50% of families be single-parent fam-
ilies?
39. The table shows the amount spent on student scholarships
(in millions of dollars) by Oberlin College in recent years.
[1995 indicates the school year 1995–96, and so on.]
*U.S. Census Bureau.
Year 1960 1970 1980 1990 2000 2003
Percent 12.8 13.2 17.5 20.8 23.2 27.5
Year 1995 1996 1997 1998 1999 2000 2001
Scholarships 19.8 22.0 25.7 27.5 28.7 31.1 34.3
150 CHAPTER 3 Functions and Graphs
(a) Use linear regression to find an equation that expresses
the amount of scholarships y as a function of the year x,
with x 0 corresponding to 1995.
(b) Assuming that the function in part (a) remains accurate,
estimate the amount spent on scholarships in 2004.
Use the following figures for Exercises 40–45. Each of the
graphs in the figure defines a function, as in Example 4.
40. State the domain and range of the function defined by
graph (a).
1
−1
−2
−3
−4
−2 −1−3
2
3
4
32
(c)
1
x
y
1
−1
−2
−3
−4
−2−3
2
3
4
342
(b)
1
x
y
−1
1
−1
−2
−3
−4
−2−3
2
3
4
32
(a)
1
x
y
−1
41. State the output (number in the range) that the function of
Exercise 40 produces from the following inputs (numbers in
the domain): 2, 1, 0, 1.
42. State the domain and range of the function defined by
graph (b).
43. State the output (number in the range) that the function of
Exercise 42 produces from the following inputs (numbers in
the domain): 2, 0, 1, 2.5, 1.5.
44. State the domain and range of the function defined by
graph (c).
45. State the output (number in the range) that the function of
Exercise 44 produces from the following inputs (numbers in
the domain): 2, 1, 0, 1/2, 1.
46. Explain why none of the graphs in the figure below defines
a function according to the procedure in Example 6. What
goes wrong?
THINKERS
47. Consider the function whose rule uses a calculator as fol-
lows: “Press COS, and then press LN; then enter a number
in the domain, and press ENTER.”* Experiment with this
function, then answer the following questions. You may not
be able to prove your answers—just make the best estimate
you can based on the evidence from your experiments.
1
−1
(a)
1
x
y
3
−3
−6
−9
−6−9
6
9
96
(b)
3
x
y
2
−2
−4
−6
−4
−1 −3
−2−6
4
6
64
(c)
2
x
y
*You don’t need to know what these keys mean to do this exercise.
(a) What is the largest set of real numbers that could be
used for the domain of this function? [If applying the
rule to a number produces an error message or a complex
number, that number cannot be in the domain.]
(b) Using the domain in part (a), what is the range of this
function?
48. Do Exercise 47 for the function whose rule is “Press 10
x
,
and then press TAN; then enter a number in the domain, and
press ENTER.”
49. The integer part function has the set of all real numbers
(written as decimals) as its domain. The rule is “For each
input number, the output is the part of the number to the left
of the decimal point.” For instance, the input 37.986 pro-
duces the output 37, and the input 1.5 produces the out-
put 1. On most calculators, the integer part function is
denoted “iPart.” On calculators that use “Intg” or “Floor”
for the greatest integer function, the integer part function is
denoted by “INT.”
(a) For each nonnegative real number input, explain why
both the integer part function and the greatest integer
function [Example 7] produce the same output.
(b) For which negative numbers do the two functions pro-
duce the same output?
(c) For which negative numbers do the two functions pro-
duce different outputs?
50. It is possible to write every even natural number uniquely as
the product of two natural numbers, one odd and one a
power of two. For example:
46 23 236 9 2
2
8 1 2
3
.
SECTION 3.2 Functional Notation 151
Consider the function whose input is the set of even integers
and whose output is the odd number you get in the above
process. So if the input is 36, the output is 9. If the input is
46, the output is 23.
(a) Write a table of values for inputs 2, 4, 6, 8, 10, 12 and 14.
(b) Find five different inputs that give an output of 3.
51. Example 11(b) showed how we create a table of values for
a function when you get to choose all the values of the
inputs. The technique presented does not work for Casio
calculators. This exercise is designed for users of Casio
calculators.
•
Enter an equation such as y x
3
2x 3 in the equa-
tion memory. This can be done by selecting TABLE in
the MAIN menu.
•
Return to the MAIN menu and select LIST. Enter the
numbers at which you want to evaluate the function as
List 1.
•
Return to the MAIN menu and select TABLE. Then
press SET-UP [that is, 2nd MENU] and select LIST
as the Variable; on the LIST menu, choose List 1.
Press EXIT and then press TABL to produce the
table.
•
Use the up/down arrow key to scroll through the table.
If you change an entry in the X column, the correspon-
ding y
1
value will automatically change.
(a) Use this technique to duplicate the table in Exam-
ple 11(b).
(b) Change the number 11 to 10, and confirm that you’ve
obtained 10
3
2(10) 3.
3.2 Functional Notation
■ Use functional notation.
■ Compute the difference quotient of a function.
■ Identify common mistakes made with functional
notation.
■ Determine the domain of a function.
■ Use a piecewise-defined function.
Functional notation is a convenient shorthand language that facilitates the analy-
sis of mathematical problems involving functions. It arises from real-life situa-
tions, such as the following.
Section Objectives
EXAMPLE 1
In Section 3.1, we saw that the 2006 Louisiana state income tax rates (for a single
person with one exemption and no deductions) were as follows:
Let I denote income, and write T(I) (read “T of I ”) to denote the amount of tax on
income I. In this shorthand language, T(7500) denotes “the tax on an income of
$7500.” The sentence “The tax on an income of $7500 is $157.50” is abbreviated
as T(7500) 157.5. Similarly, T(25,000) 693.75 says that the tax on an income
of $25,000 is $693.75. There is nothing that forces us to use the letters T and I here:
Any choice of letters will do, provided that we
make clear what is meant by these letters. ■
EXAMPLE 2
Recall that a falling rock travels 16t
2
feet after t seconds. Let d(t) stand for the
phrase “the distance the rock has traveled after t seconds.” Then the sentence “The
distance the rock has traveled after t seconds is 16t
2
feet” can be abbreviated as
d(t) 16t
2
. For instance,
d(1) 16
1
2
16
means “the distance the rock has traveled after 1 second is 16 feet,” and
d(4) 16
4
2
256
means “the distance the rock has traveled after 4 seconds is 256 feet.” ■
Functional notation is easily adapted to mathematical settings, in which the
particulars of time, distance, etc., are not mentioned. Suppose a function is given.
Denote the function by f and let x denote a number in the domain. Then
f(x) denotes the output produced by input x.
For example, f (6) is the output produced by the input 6. The sentence
“y is the output produced by input x according to the rule of the function f”
is abbreviated
y f (x),
which is read “y equals f of x.” The output f (x) is sometimes called the value of
the function f at x.
In actual practice, functions are seldom presented in the style of domain,
rule, range, as they have been here. Usually, you will be given a phrase such
152 CHAPTER 3 Functions and Graphs
Income
At Least But Less Than Tax
0 $12,500 2.1%
$12,500 $25,000 $262.50 3.45% of amount over $12,500
$25,000 $693.75 4.8% of amount over $25,000
CAUTION
The parentheses in d(t ) do not denote
multiplication as in the algebraic
equation 3(a b) 3a 3b. The
entire symbol d(t ) is part of a
shorthand language. In particular
d(1 4) is not equal to d(1) d (4).
We saw above that d(1) 16 and
d(4) 256, so d(1) d(4)
16 256 272. But d(1 4) is
“the distance traveled after 1 4
seconds,” that is, the distance after
5 seconds, namely, 16
5
2
400.
In general,
Functional notation is a convenient
shorthand for phrases and
sentences in the English language.
It is not the same as ordinary
algebraic notation.
as “the function f (x)
x
2
1
.” This should be understood as a set of
directions:
Name of function Input number
f(x)
x
2
1
123 14243
Output number Directions that tell you what to do with
input x to produce the corresponding output
f(x), namely, “square it, add 1, and take the
square root of the result.”
For example, to find f (3), the output of the function f for input 3, simply replace x
by 3 in the formula:
f (x)
x
2
1
f (3)
3
2
1
10.
Similarly, replacing x by 5 and 0 shows that
f(5)
(5)
2
1
26 and f(0)
0
2
1
1.
EXAMPLE 3
The expression
h(x)
x
x
2
1
5
defines the function h whose rule is
For input x, the output is the number
x
x
2
1
5
.
Find each of the following:
h(3
), h(2), h(a), h(r
2
3), h(c 2
).
SOLUTION To find h(3
) and h(2), replace x by 3
and 2, respectively,
in the rule of h:
h(3
)
(
3
3
)
2
1
5
3
8
1
and h(2)
(
2
2
)
2
1
5
3.
The value of the function h at any quantity, such as a, r
2
3, etc., can be found
by using the same procedure: Replace x in the formula for h(x) by that quantity:
h(a)
(
a
a
)
2
1
5
a
2
a
5
1
h(r
2
3)
(
(
r
r
2
2
3
3
)
)
2
1
5
r
4
6
r
r
2
2
2
9 5
r
4
r
2
6r
2
2
14
h(c 2
)
(
c
c
2
2
)
2
1
5
c
c
2
2
5
1
c
c
2
7
1
. ■
SECTION 3.2 Functional Notation 153
———
———
———
———
TECHNOLOGY TIP
Functional notation can be used on
calculators other than the Casio. If the
function is y
1
in the function memory,
then
y
1
(5) ENTER
gives the value of the function at
x 5.
You can key in y
1
from the
keyboard on TI-86/89 and HP-39gs.
On TI-84+, y
1
is in the FUNCTION
submenu of the Y-VARS menu.
Warning: Keying in y
1
(5) on Casio
will produce an answer, but it usually
will be wrong.
When functional notation is used in expressions such as f (x) or f (x h),
the same basic rule applies: Replace x in the formula by the entire expression in
parentheses.
EXAMPLE 4
If f (x) x
2
x 2, then
f (3) (3)
2
(3) 2 4
f (3) (3
2
3 2) 10
f (x) (x)
2
(x) 2 x
2
x 2.
Note that for this function, f (x) is not the same as f (x), because f (x) is the
negative of the number f (x), that is,
f (x) (x
2
x 2) x
2
x 2. ■
EXAMPLE 5
If f (x) x
2
x 2 and h 0, find
(a) f (x h) (b) f (x h) f (x)(c)
f (x h
h
) f (x)
.
SOLUTION
(a) Replace x by x h in the rule of the function:
f (x h) (x h)
2
(x h) 2
x
2
2xh h
2
x h 2.
(b) By part (a),
f(x h) f(x) [(x h)
2
(x h) 2] [x
2
x 2]
[x
2
2xh h
2
x h 2] [x
2
x 2]
x
2
2xh h
2
x h 2 x
2
x 2
2xh h
2
h.
(c) By part (b), we have
f(x h
h
) f(x)
2xh
h
h
2
h
h(2x
h
h 1)
2x h 1. ■
If f is a function, then the quantity
f (x h
h
) f (x)
, as in Example 5(c), is
called the difference quotient of f. Difference quotients, whose significance is
explained in Section 3.6, play an important role in calculus.
154 CHAPTER 3 Functions and Graphs
TECHNOLOGY TIP
One way to evaluate a function f (x) is
to enter its rule as an equation
y f (x) in the equation memory and
use TABLE or (on TI-86) EVAL; see
Example 11 in Section 3.1.
EXAMPLE 6
Compute and simplify the difference quotient for the function f (x) x
3
2x.
SOLUTION
f (x h) f(x)
64444744448 64748
f (x h
h
) f (x)
3x
2
3xh h
2
2. ■
As the preceding examples illustrate, functional notation is a specialized short-
hand language. Treating it as ordinary algebraic notation may lead to mistakes.
EXAMPLE 7
Many students make untrue assumptions when working with functional notation.
Here are examples of three of the items listed in the Caution box.
If f (x) x
2
, then
f (3 2) f (5) 5
2
25.
But
f (3) f (2) 3
2
2
2
9 4 13.
So f (3 2) f (3) f (2).
If f (x) x 7, then
f (3
4) f (12) 12 7 19.
But
f (3) f (4) (3 7)(4 7) 10
11 110.
So f (3
4) f (3) f (4).
If f(x) x
2
1, then
f (2
3) (2
3)
2
1 36 1 37.
But
f (2)
3 (2
2
1)3 5
3 15.
So f (2
3) f (2)
3. ■
h(3x
2
3xh h
2
2)
h
3x
2
h 3xh
2
h
3
2h
h
x
3
3x
2
h 3xh
2
h
3
2x 2h x
3
2x
h
[x
3
3x
2
h 3xh
2
h
3
2x 2h] [x
3
2x]
h
[(x h)
3
2(x h)] [x
3
2x]
h
SECTION 3.2 Functional Notation 155
CAUTION
It is common to make mistakes with
functional notation.
Remember that, in general:
f (a b) f (a) f (b)
f (a b) f (a) f (b)
f (ab) f (a)f (b)
f (ab) af (b)
f (ab) f (a)b