1-3 Functions
48. Price and Revenue. Refer to Problem 46. The revenue
from the sale of x cases of paper at $y per case is given by
the product R ⫽ xy.
(A) Use the results from Problem 46 to complete the
following table of revenues.
y
35
40
45
R
27
(B) Does the revenue increase or decrease if the price is
increased from $40 to $45? By how much?
(C) Does the revenue increase or decrease if the price is
decreased from $40 to $35? By how much?
(D) If the current price of paper is $40 per case and the
company wants to increase revenue, should they raise
the price $5, lower the price $5, or leave the price
unchanged?
Section 1-3 Functions
Definition of a Function
Functions Defined by Equations
Function Notation
Application
A Brief History of the Function Concept
The idea of correspondence plays a central role in the formulation of the function concept. You have already had experiences with correspondences in everyday life. For example:
To each person there corresponds an age.
To each item in a store there corresponds a price.
To each automobile there corresponds a license number.
To each circle there corresponds an area.
To each number there corresponds its cube.
One of the most important aspects of any science (managerial, life, social,
physical, computer, etc.) is the establishment of correspondences among various
types of phenomena. Once a correspondence is known, predictions can be made.
A chemist can use a gas law to predict the pressure of an enclosed gas, given its
temperature. An engineer can use a formula to predict the deflections of a beam
subject to different loads. A computer scientist can use formulas to compare the
efficiency of algorithms for sorting data stored in a computer. An economist would
like to be able to predict interest rates, given the rate of change of the money
supply. And so on.
Definition of a Function
What do all the preceding examples have in common? Each describes the matching of elements from one set with elements in a second set. Consider Tables 1–3,
which list values for the cube, square, and square root, respectively.
28
1 FUNCTIONS AND GRAPHS
T A B L E
1
T A B L E
Domain
(number)
Range
(cube)
⫺2
⫺1
0
1
2
⫺8
⫺1
0
1
8
Domain
(number)
⫺2
⫺1
0
1
2
2
T A B L E
Range
(square)
Domain
(number)
0
4
1
0
1
4
9
3
Range
(square root)
0
1
⫺1
2
⫺2
3
⫺3
Tables 1 and 2 specify functions, but Table 3 does not. Why not? The definition of the term function will explain.
DEFINITION
1
RULE FORM OF THE DEFINITION OF A FUNCTION
A function is a rule that produces a correspondence between two sets of
elements such that to each element in the first set there corresponds one
and only one element in the second set.
The first set is called the domain and the set of all corresponding elements in the second set is called the range.
Tables 1 and 2 specify functions, since to each domain value there corresponds
exactly one range value (for example, the cube of ⫺2 is ⫺8 and no other number). On the other hand, Table 3 does not specify a function, since to at least one
domain value there corresponds more than one range value (for example, to the
domain value 9 there corresponds ⫺3 and 3, both square roots of 9).
Remark
Explore/Discuss
1
Some graphing utilities use the term range to refer to the window variables. In
this book, range will always refer to the range of a function.
Consider the set of students enrolled in a college and the set of faculty
members of that college. Define a correspondence between the two sets
by saying that a student corresponds to a faculty member if the student is
currently enrolled in a course taught by the faculty member. Is this correspondence a function? Discuss.
Since a function is a rule that pairs each element in the domain with a corresponding element in the range, this correspondence can be illustrated by using
ordered pairs of elements, where the first component represents a domain element
1-3 Functions
29
and the second component represents the corresponding range element. Thus, the
functions defined in Tables 1 and 2 can be written as follows:
Function 1 ⫽ {(⫺2, ⫺8), (⫺1, ⫺1), (0, 0), (1, 1), (2, 8)}
Function 2 ⫽ {(⫺2, 4), (⫺1, 1), (0, 0), (1, 1), (2, 4)}
In both cases, notice that no two ordered pairs have the same first component
and different second components. On the other hand, if we list the set A of ordered
pairs determined by Table 3, we have
A ⫽ {(0, 0), (1, 1), (1, ⫺1), (4, 2), (4, ⫺2), (9, 3), (9, ⫺3)}
In this case, there are ordered pairs with the same first component and different
second components; for example, (1, 1) and (1, ⫺1) both belong to the set A.
Once again, we see that Table 3 does not define a function.
This suggests an alternative but equivalent way of defining functions that produces additional insight into this concept.
DEFINITION
2
EXAMPLE
1
SET FORM OF THE DEFINITION OF A FUNCTION
A function is a set of ordered pairs with the property that no two
ordered pairs have the same first component and different second
components.
The set of all first components in a function is called the domain of the
function, and the set of all second components is called the range.
Functions Defined as Sets of Ordered Pairs
(A) The set S ⫽ {(1, 4), (2, 3), (3, 2), (4, 3), (5, 4)} defines a function since
no two ordered pairs have the same first component and different second
components. The domain and range are
Domain ⫽ {1, 2, 3, 4, 5}
Range ⫽ {2, 3, 4}
Set of first components
Set of second components
(B) The set T ⫽ {(1, 4), (2, 3), (3, 2), (2, 4), (1, 5)} does not define a function since there are ordered pairs with the same first component and different second components [for example, (1, 4) and (1, 5)].
MATCHED PROBLEM
1
Determine whether each set defines a function. If it does, then state the domain
and range.
(A) S ⫽ {(⫺2, 1), (⫺1, 2), (0, 0), (⫺1, 1), (⫺2, 2)}
(B) T ⫽ {(⫺2, 1), (⫺1, 2), (0, 0), (1, 2), (2, 1)}
30
1 FUNCTIONS AND GRAPHS
Functions Defined by Equations
Both versions of the definition of a function are quite general, with no restrictions
on the type of elements that make up the domain or range. In this text, unless
otherwise indicated, the domain and range of a function will be sets of real
numbers.
Defining a function by displaying the rule of correspondence in a table or listing all the ordered pairs in the function only works if the domain and range are
relatively small finite sets. Functions with finite domains and ranges are used
extensively in certain specialized areas, such as computer science, but most applications of functions involve infinite domains and ranges. If the domain and range
of a function are infinite sets, then the rule of correspondence cannot be displayed
in a table, and it is not possible to actually list all the ordered pairs belonging to
the function. For most functions, we use an equation in two variables to specify
both the rule of correspondence and the set of ordered pairs.
Consider the equation
y ⫽ x2 ⫹ 2x
x any real number
(1)
This equation assigns to each domain value x exactly one range value y. For
example,
If x ⫽ 4,
then
y ⫽ (4)2 ⫹ 2(4) ⫽ 24
If x ⫽ ⫺ 13,
then
y ⫽ (⫺ 13)2 ⫹ 2(⫺ 13) ⫽ ⫺ 59
Thus, we can view equation (1) as a function with rule of correspondence
y ⫽ x2 ⫹ 2x
x2 ⫹ 2x corresponds to x
or, equivalently, as a function with set of ordered pairs
ⱍ
{(x, y) y ⫽ x2 ⫹ 2x, x a real number}
The variable x is called an independent variable, indicating that values can be
assigned “independently” to x from the domain. The variable y is called a dependent variable, indicating that the value of y “depends” on the value assigned to x
and on the given equation. In general, any variable used as a placeholder for
domain values is called an independent variable; any variable used as a placeholder for range values is called a dependent variable.
Which equations can be used to define functions?
FUNCTIONS DEFINED BY EQUATIONS
In an equation in two variables, if to each value of the independent variable there corresponds exactly one value of the dependent variable, then
the equation defines a function.
If there is any value of the independent variable to which there corresponds more than one value of the dependent variable, then the equation
does not define a function.
1-3 Functions
31
Notice that we have used the phrase “an equation defines a function” rather
than “an equation is a function.” This is a somewhat technical distinction, but it
is employed consistently in mathematical literature and we will adhere to it in
this text.
(A) Graph y ⫽ x2 ⫺ 4 for ⫺5 ⱕ x ⱕ 5 and ⫺5 ⱕ y ⱕ 5 and trace
along this graph. Discuss the relationship between the coordinates
displayed while tracing and the function defined by this equation.
(B) The graph of the equation x2 ⫹ y2 ⫽ 16 is a circle. Since most
graphing utilities will accept only equations that have been solved
for y, we must graph the equations y1 ⫽ 兹16 ⫺ x2 and
y2 ⫽ ⫺ 兹16 ⫺ x2 to produce a graph of the circle. Graph these
equations for ⫺5 ⱕ x ⱕ 5 and ⫺5 ⱕ y ⱕ 5. Then try different
values for Xmin and Xmax until the graph looks more like a circle.
Use the trace feature to find two points on this circle with the same
x coordinate and different y coordinates.
(C) Is it possible to graph a single equation of the form y ⫽ (expression
in x) on your graphing utility and obtain a graph that is not the
graph of a function? Explain your answer.
Explore/Discuss
2
Remark
If we want the graph of a circle to actually appear to be circular, we must choose
window variables so that a unit length on the x axis is the same number of pixels
as a unit length on the y axis. Such a window is often referred to as a squared
viewing window. Most graphing utilities have an option under the zoom menu
that will do this automatically.
Not all equations determine functions. One way to determine if an equation
does determine a function is to examine its graph. The graphs of the equations
y ⫽ x2 ⫺ 4
and
x2 ⫹ y2 ⫽ 16
are shown in Figure 1.
FIGURE 1
y
Graphs of equations and the
vertical line test.
y
5
5
x
⫺5
5
x
⫺5
5
y ⫽ x2 ⫺ 4
x 2 ⫹ y 2 ⫽ 16
⫺5
(a)
⫺5
(b)
The graph in Figure 1(a) is a parabola and the graph in Figure 1(b) is a circle. Any vertical line intersects the parabola in exactly one point. This shows that
each value of the independent variable x corresponds to exactly one value of the
dependent variable y. Thus, the equation y ⫽ x2 ⫺ 4 defines a function. On the
32
1 FUNCTIONS AND GRAPHS
other hand, there are vertical lines that intersect the circle in Figure 1(b) in two
points. This indicates that there are values of the independent variable x that correspond to two different values of the dependent variable y. Consequently, the
equation x2 ⫹ y2 ⫽ 16 does not define a function. These observations are generalized in Theorem 1.
THEOREM
1
VERTICAL LINE TEST FOR A FUNCTION
An equation defines a function if each vertical line in the rectangular
coordinate system passes through at most one point on the graph of the
equation.
If any vertical line passes through two or more points on the graph of an
equation, then the equation does not define a function.
Since the expression x2 ⫺ 4 represents a real number for all real values of x,
the function defined by the equation y ⫽ x2 ⫺ 4 is defined for all real numbers.
Thus, its domain is the set of all real numbers, often denoted by the letter R
or the interval* (⫺⬁, ⬁). On the other hand, the expression 兹16 ⫺ x2 represents
a real number only if 16 ⫺ x2 ⱖ 0. This inequality is equivalent to x2 ⱕ 16 or
⫺4 ⱕ x ⱕ 4. Thus, the domain of the function y ⫽ 兹16 ⫺ x2 is
{x ⫺4 ⱕ x ⱕ 4} or [⫺4, 4]. Unless stated to the contrary, we will adhere to the
following convention regarding domains and ranges for functions defined by
equations.
ⱍ
AGREEMENT ON DOMAINS AND RANGES
If a function is defined by an equation and the domain is not indicated,
then we assume that the domain is the set of all real number replacements of the independent variable that produce real values for the
dependent variable. The range is the set of all values of the dependent
variable corresponding to these domain values.
EXAMPLE
2
Solution
Finding the Domain of a Function
Find the domain of the function defined by the equation y ⫽ 4 ⫹ 兹x, assuming x is the independent variable.
For 兹x to be real, x must be greater than or equal to 0. Thus,
Domain:
ⱍ
{x x ⱖ 0}
or [0, ⬁)
Note that in many cases we will dispense with set notation and simply write
x ⱖ 0 instead of {x x ⱖ 0}.
ⱍ
*See Appendix A, Section A-8, for a discussion of interval notation.
1-3 Functions
MATCHED PROBLEM
2
33
Find the domain of the function defined by the equation y ⫽ 3 ⫹ 兹⫺x, assuming x is the independent variable.
Function Notation
We will use letters to name functions and to provide a very important and
convenient notation for defining functions. For example, if f is the name of the
function defined by the equation y ⫽ 2x ⫹ 1, then instead of the more formal
representations
f : y ⫽ 2x ⫹ 1
Rule of correspondence
or
ⱍ
f : {(x, y) y ⫽ 2x ⫹ 1}
Set of ordered pairs
we simply write
f(x) ⫽ 2x ⫹ 1
Function notation
The symbol f(x) is read “f of x,” “f at x,” or “the value of f at x” and represents
the number in the range of the function f to which the domain value x is paired.
Thus, f(3) is the range value for the function f associated with the domain value
3. We find this range value by replacing x with 3 wherever x occurs in the function definition
f(x) ⫽ 2x ⫹ 1
and evaluating the right side,
f(3) ⫽ 2 ⴢ 3 ⫹ 1
⫽6⫹1
⫽7
The statement f(3) ⫽ 7 indicates in a concise way that the function f assigns the
range value 7 to the domain value 3 or, equivalently, that the ordered pair (3, 7)
belongs to f.
The symbol f : x → f(x), read “f maps x into f(x),” is also used to denote the
relationship between the domain value x and the range value f(x) (see Fig. 2).
Whenever we write y ⫽ f(x), we assume that x is an independent variable and
that y and f(x) both represent the dependent variable.
FIGURE 2
Function notation.
f
x
DOMAIN
f (x)
RANGE
The function f “maps” the domain
value x into the range value f (x).
34
1 FUNCTIONS AND GRAPHS
Letters other than f and x can be used to represent functions and independent
variables. For example,
g(t) ⫽ t 2 ⫺ 3t ⫹ 7
defines g as a function of the independent variable t. To find g(⫺2), we replace
t by ⫺2 wherever t occurs in
g(t) ⫽ t 2 ⫺ 3t ⫹ 7
and evaluate the right side:
g(ⴚ2) ⫽ (ⴚ2)2 ⫺ 3(ⴚ2) ⫹ 7
⫽4⫹6⫹7
⫽ 17
Thus, the function g assigns the range value 17 to the domain value ⫺2; the
ordered pair (⫺2, 17) belongs to g.
It is important to understand and remember the definition of the symbol f(x):
DEFINITION
3
EXAMPLE
3
THE SYMBOL f(x)
The symbol f(x) represents the real number in the range of the function f
corresponding to the domain value x. Symbolically, f : x → f(x). The
ordered pair (x, f(x)) belongs to the function f. If x is a real number that
is not in the domain of f, then f is not defined at x and f(x) does not
exist.
Evaluating Functions
For
15
x⫺3
f(x) ⫽
find:
(A) f(6)
Solutions
(A) f(6)
⫽
g(x) ⫽ 16 ⫹ 3x ⫺ x2
(B) g(⫺7)
(C) h(⫺2)
h(x) ⫽
6
x
兹 ⫺1
(D) f(0) ⫹ g(4) ⫺ h(16)
15 *
15
⫽
⫽5
6⫺3
3
(B) g(ⴚ7) ⫽ 16 ⫹ 3(ⴚ7) ⫺ (ⴚ7)2 ⫽ 16 ⫺ 21 ⫺ 49 ⫽ ⫺54
(C) h(⫺2) ⫽
6
兹⫺2 ⫺ 1
*Throughout the book, dashed boxes—called think boxes—are used to represent steps that are usually
performed mentally.
1-3 Functions
35
But 兹⫺2 is not a real number. Since we have agreed to restrict the domain
of a function to values of x that produce real values for the function, ⫺2 is
not in the domain of h and h(⫺2) is not defined.
(D) f(0) ⫹ g(4) ⫺ h(16)
⫽
6
15
⫹ [16 ⫹ 3(4) ⫺ 42] ⫺
0⫺3
16
兹 ⫺1
⫽
15
6
⫹ 12 ⫺
⫺3
3
⫽ ⫺5 ⫹ 12 ⫺ 2 ⫽ 5
MATCHED PROBLEM
3
EXAMPLE
4
Use the functions in Example 3 to find
(A) f(⫺2)
(B) g(6)
(D)
f(8)
h(9)
Finding Domains of Functions
Find the domains of functions f, g, and h:
f(x) ⫽
Solution
(C) h(⫺8)
15
x⫺3
g(x) ⫽ 16 ⫹ 3x ⫺ x2
h(x) ⫽
6
兹x ⫺ 1
Domain of f
The fraction 15/(x ⫺ 3) represents a real number for all replacements of x by real
numbers except x ⫽ 3, since division by 0 is not defined. Thus, f(3) does not
exist, and the domain of f is the set of all real numbers except 3. That is,
ⱍ
Domain of f ⫽ {x x ⫽ 3)
⫽ (⫺⬁, 3) 傼 (3, ⬁)
Set notation
Interval notation
We often simplify this by writing
f(x) ⫽
15
x⫺3
x⫽3
Domain of g
The domain is R, the set of all real numbers, since 16 ⫹ 3x ⫺ x2 represents a
real number for all replacements of x by real numbers. To express this domain in
interval notation, we write
Domain of g ⫽ (⫺⬁, ⬁)
Domain of h
Since 兹x is not a real number for negative real numbers x, x must be a nonnegative real number. But since 兹1 ⫽ 1, evaluating h(1) would result in division
36
1 FUNCTIONS AND GRAPHS
by 0. Thus, the domain of h is all nonnegative real numbers except 1. This can
be written as
ⱍ
Domain of h ⫽ {x x ⱖ 0, x ⫽ 1}
⫽ [0, 1) 傼 (1, ⬁)
Set notation
Interval notation
or, more informally, as
h(x) ⫽
6
兹x ⫺ 1
x ⱖ 0, x ⫽ 1
Find the domains of functions F, G, and H:
MATCHED PROBLEM
4
F(x) ⫽ x2 ⫹ 5x ⫺ 2
Explore/Discuss
G(x) ⫽
兹x
x⫺3
H(x) ⫽
x
1
⫹
x x⫹2
Let x and h be any real numbers.
(A) If f(x) ⫽ 3x ⫹ 2, which of the following is correct?
(i) f(x ⫹ h) ⫽ 3x ⫹ 2 ⫹ h
(ii) f(x ⫹ h) ⫽ 3x ⫹ 3h ⫹ 2
(iii) f(x ⫹ h) ⫽ 3x ⫹ 3h ⫹ 4
(B) If f(x) ⫽ x2, which of the following is correct?
(i) f(x ⫹ h) ⫽ x2 ⫹ h2
(ii) f(x ⫹ h) ⫽ x2 ⫹ h
(iii) f(x ⫹ h) ⫽ x2 ⫹ 2xh ⫹ h2
(C) If f(x) ⫽ x2 ⫹ 3x ⫹ 2, write a verbal description of the operations that
must be performed to evaluate f(x ⫹ h).
3
In addition to evaluating functions at specific numbers, it is important to be
able to evaluate functions at expressions that involve one or more variables. For
example, the difference quotient
f(x ⫹ h) ⫺ f(x)
h
x and x ⫹ h in the domain of f, h ⫽ 0
is studied extensively in a calculus course.
EXAMPLE
5
Evaluating and Simplifying a Difference Quotient
*
For f(x) ⫽ x2 ⫹ 4x ⫹ 5, find and simplify:
f(x ⫹ h) ⫺ f(x)
,h⫽0
(A) f(x ⫹ h)
(B)
h
*The symbol
denotes problems that are related to calculus.
1-3 Functions
Solutions
37
(A) To find f(x ⫹ h), we replace x with x ⫹ h everywhere it appears in the equation that defines f and simplify:
f(x ⴙ h) ⫽ (x ⴙ h)2 ⫹ 4(x ⴙ h) ⫹ 5
⫽ x2 ⫹ 2xh ⫹ h2 ⫹ 4x ⫹ 4h ⫹ 5
(B) Using the result of part (A), we get
f(x ⫹ h) ⫺ f(x) x2 ⫹ 2xh ⫹ h2 ⫹ 4x ⫹ 4h ⫹ 5 ⫺ (x2 ⫹ 4x ⫹ 5)
⫽
h
h
MATCHED PROBLEM
⫽
x2 ⫹ 2xh ⫹ h2 ⫹ 4x ⫹ 4h ⫹ 5 ⫺ x2 ⫺ 4x ⫺ 5
h
⫽
2xh ⫹ h2 ⫹ 4h
h
⫽
h(2x ⫹ h ⫹ 4)
⫽ 2x ⫹ h ⫹ 4
h
Repeat Example 5 for f(x) ⫽ x2 ⫹ 3x ⫹ 7.
5
CAUTION
1. If f is a function, then the symbol f(x ⫹ h) represents the value of f at
the number x ⫹ h and must be evaluated by replacing the independent
variable in the equation that defines f with the expression x ⫹ h, as
we did in Example 5. Do not confuse this notation with the familiar
algebraic notation for multiplication:
f(x ⫹ h) ⫽ fx ⫹ fh
f(x ⫹ h) is function notation.
4(x ⫹ h) ⫽ 4x ⫹ 4h
4(x ⫹ h) is algebraic multiplication
notation.
2. There is another common incorrect interpretation of the symbol
f(x ⫹ h). If f is an arbitrary function, then
f(x ⫹ h) ⫽ f(x) ⫹ f(h)
It is possible to find some particular functions for which f(x ⫹ h) ⫽
f(x) ⫹ f(h) is a true statement, but in general these two expressions
are not equal.
Application
The next example explores the relationship between the algebraic definition of a
function, the numeric values of the function, and a graphic representation of the
function. The interplay between the algebraic, numeric, and graphic aspects of a
function is one of the central themes of this book. In this example, we also see
how a function can be used to describe data from the real world, a process that
is generally referred to as mathematical modeling.
38
1 FUNCTIONS AND GRAPHS
EXAMPLE
Consumer Debt
6
Revolving-credit debt (in billions of dollars) in the United States over a 20year period is given in Table 4. A financial analyst used statistical techniques
to produce a mathematical model for this data:
T A B L E 4 Revolving-Credit Debt
Year
f(x) ⫽ 0.62x2 ⫺ x ⫹ 5.1
Total Debt
(Billions)
1970
1975
1980
1985
1990
where x ⫽ 0 corresponds to 1970.
(A) To compare the data in Table 4 and the values produced by the modeling function f, use a graphing utility to complete Table 5.
$5.1
$15.0
$58.5
$128.9
$234.8
T A B L E
x
Debt
f(x)
Source: Federal Reserve System.
0
5.1
5
5
15.0
10
58.5
15
128.9
20
234.8
(B) Sketch by hand the graph of the modeling function f and the original data
using the same axes.
(C) Use the modeling function f to estimate the debt to the nearest tenth of
a billion in 1988 and in 1992.
Solutions
FIGURE 3
(A) As we mentioned earlier, most graphing utilities have a built-in routine for
computing a table of values (Fig. 3). If yours does not, then simply evaluate the function at each value of x given in the table.
x
Debt
f(x)
0
5.1
5.1
5
15.0
15.6
10
58.5
57.1
15
128.9
129.6
20
234.8
233.1
(B) Figure 4 shows a sketch of the graph of y ⫽ f(x) and the original data points
in Table 4 with 0 corresponding to 1970.
FIGURE 4
y
y ⫽ f (x) ⫽ 0.62x 2 ⫺ x ⫹ 5.1
250
200
150
100
50
5
10
15
20
25
x
(C) Evaluate f(x) at 18 and at 22:
f(18) ⫽ 188.0
f(22) ⫽ 283.2.
1-3 Functions
39
Thus, the revolving-credit debt should be $188 billion in 1988 and $283.2
billion in 1992.
MATCHED PROBLEM
6
Credit union debt (in billions of dollars) in the United States is given in Table 6.
Repeat Example 6 using this data and the modeling function.
y ⫽ f(x) ⫽ 0.5x2 ⫹ 5.6x ⫹ 46.6
T A B L E
Year
1970
1975
1980
1985
1990
6 Credit Union Debt
Total Debt
(Billions)
$48.7
$82.9
$147.0
$245.1
$347.1
Source: Federal Reserve System.
Remarks
1. Modeling functions like the function f in Example 6 provide reasonable and
useful representations of the given data, but they do not always correctly predict future behavior. For example, the model in Example 6 indicated that the
revolving-credit debt in 1992 should be about $283.2 billion. But the actual
debt for 1992 turned out to be $267.9 billion, which differs from the predicted value by over $18 billion. Proper use of mathematical models requires
both an understanding of the techniques used to develop the model and frequent reevaluation, modification, and interpretation of the results produced by
the model.
2. Later in this chapter we will discuss methods for finding a function f that
models a given set of data. It turns out that this is easy to do with a graphing utility.
A Brief History of the Function Concept
The history of the use of functions in mathematics illustrates the tendency of
mathematicians to extend and generalize a concept. The word “function” appears
to have been first used by Leibniz in 1694 to stand for any quantity associated
with a curve. By 1718, Johann Bernoulli considered a function any expression
made up of constants and a variable. Later in the same century, Euler came to
regard a function as any equation made up of constants and variables. Euler made
extensive use of the extremely important notation f(x), although its origin is generally attributed to Clairaut (1734).
The form of the definition of function that has been used until well into this
century (many texts still contain this definition) was formulated by Dirichlet
(1805–1859). He stated that, if two variables x and y are so related that for
each value of x there corresponds exactly one value of y, then y is said to be a
40
1 FUNCTIONS AND GRAPHS
(single-valued) function of x. He called x, the variable to which values are
assigned at will, the independent variable, and y, the variable whose values depend
on the values assigned to x, the dependent variable. He called the values assumed
by x the domain of the function, and the corresponding values assumed by y the
range of the function.
Now, since set concepts permeate almost all mathematics, we have the more
general definition of function presented in this section in terms of sets of ordered
pairs of elements.
Answers to Matched Problems
1. (A) S does not define a function.
(B) T defines a function with domain {⫺2, ⫺1, 0, 1, 2} and range {0, 1, 2}.
2. x ⱕ 0
3. (A) ⫺3
(B) ⫺2
(C) Does not exist
(D) 1
4. Domain of F: all real numbers
Domain of G: x ⱖ 0, x ⫽ 3 or [0, 3) 艛 (3, ⬁)
Domain of H: all real numbers except 0 and ⫺2 or (⫺⬁, ⫺2) 艛 (⫺2, 0) 艛 (0, 00)
5. (A) x2 ⫹ 2xh ⫹ h2 ⫹ 3x ⫹ 3h ⫹ 7
(B) 2x ⫹ h ⫹ 3
(C) $309.4 billion;
6. (A) x
(B)
y
5
10
15
20
0
$411.8 billion
y ⫽ f (x) ⫽ 0.5x 2 ⫹ 5.6x ⫹ 46.6
Debt 48.7 82.9 147
245.1 347.1
400
f (x)
46.6
87.1
152.6
243.1
358.6
300
200
100
5
EXERCISE 1-3
5.
Indicate whether each table in Problems 1–6 defines a
function.
Domain
⫺1
0
1
Range 2. Domain
1
2
3
2
4
6
15
Domain
⫺1
0
1
2
A
1.
10
20
25
x
Range 6. Domain
3
2
3
4
5
Domain
1
3
5
Range 4. Domain
3
5
7
9
⫺1
⫺2
⫺3
8
9
Range
1
3
5
Indicate whether each set in Problems 7–12 defines a function.
Find the domain and range of each function.
7. {(2, 4), (3, 6), (4, 8), (5, 10)}
3.
Range
Range
0
5
8
8. {(⫺1, 4), (0, 3), (1, 2), (2, 1)}
9. {(10, ⫺10), (5, ⫺5), (0, 0), (5, 5) (10, 10)}
10. {(⫺10, 10), (⫺5, 5), (0, 0), (5, 5), (10, 10)}
11. {(0, 1), (1, 1), (2, 1), (3, 2), (4, 2), (5, 2)}
12. {(1, 1), (2, 1), (3, 1), (1, 2), (2, 2), (3, 2)}
1-3 Functions
Indicate whether each graph in Problems 13–18 is the graph
of a function.
18.
41
y
10
13.
y
10
x
⫺10
10
x
⫺10
10
⫺10
Problems 19–30 refer to the functions
⫺10
14.
f(x) ⫽ 3x ⫺ 5
y
F(m) ⫽ 3m ⫹ 2m ⫺ 4
2
g(t) ⫽ 4 ⫺ t
G(u) ⫽ u ⫺ u2
10
Evaluate as indicated.
x
⫺10
10
⫺10
15.
y
19. f(⫺1)
20. g(6)
21. G(⫺2)
22. F(⫺3)
23. F(⫺1) ⫹ f(3)
24. G(2) ⫺ g(⫺3)
25. 2F(⫺2) ⫺ G(⫺1)
26. 3G(⫺2) ⫹ 2F(⫺1)
27.
f (0) ⴢ g(⫺2)
F(⫺3)
28.
g(4) ⴢ f (2)
G(1)
29.
f (4) ⫺ f (2)
2
30.
g(5) ⫺ g(3)
2
10
x
⫺10
10
In Problems 31–42, find the domain of the indicated function.
Express answers informally and formally using interval
notation.
⫺10
16.
B
31. f(x) ⫽ 4 ⫺ 9x ⫹ 3x2
y
10
33. f (x) ⫽
x
⫺10
10
y
34. f (x) ⫽
x
x⫺3
35. f (x) ⫽ 2 ⫺ 3兹x
36. f (x) ⫽ 4兹x ⫹ 3
37. f (x) ⫽ 5 ⫹ 兹⫺x
38. f (x) ⫽ 2兹⫺x ⫺ 1
39. f (x) ⫽
1
1
⫹
x⫹1 x⫺1
40. f (x) ⫽
x
3
⫺
x⫺2 x⫹3
41. f (x) ⫽
兹x
x⫺5
42. f (x) ⫽
兹⫺x
x⫹4
⫺10
17.
2
4⫺x
32. f(x) ⫽ 1 ⫹ 7x ⫺ 5x2
10
x
⫺10
10
⫺10
In Problems 43–46, find a function f that makes all three
equations true. [Hint: There may be more than one possible
answer, but there is one obvious answer suggested by the
pattern illustrated in the equations.]
43. f(1) ⫽ 2(1) ⫺ 3
f(2) ⫽ 2(2) ⫺ 3
f(3) ⫽ 2(3) ⫺ 3
44. f(1) ⫽ 5(1)2 ⫺ 6
f(2) ⫽ 5(2)2 ⫺ 6
f(3) ⫽ 5(3)2 ⫺ 6
42
1 FUNCTIONS AND GRAPHS
45. f(1) ⫽ 4(1)2 ⫺ 2(1) ⫹ 9
f(2) ⫽ 4(2)2 ⫺ 2(2) ⫹ 9
f(3) ⫽ 4(3)2 ⫺ 2(3) ⫹ 9
47. If F(s) ⫽ 3s ⫹ 15, find
46. f(1) ⫽ ⫺8 ⫹ 5(1) ⫺ 2(1)2
f(2) ⫽ ⫺8 ⫹ 5(2) ⫺ 2(2)2
f(3) ⫽ ⫺8 ⫹ 5(3) ⫺ 2(3)2
F(2 ⫹ h) ⫺ F(2)
.
h
In Problems 65–72, find and simplify:
(A)
f (x ⫹ h) ⫺ f (x)
h
(B)
65. f(x) ⫽ 3x ⫺ 4
66. f(x) ⫽ ⫺2x ⫹ 5
67. f(x) ⫽ x ⫺ 1
68. f(x) ⫽ x2 ⫹ x ⫺ 1
69. f(x) ⫽ ⫺3x2 ⫹ 9x ⫺ 12
70. f(x) ⫽ ⫺x2 ⫺ 2x ⫺ 4
71. f(x) ⫽ x3
72. f(x) ⫽ x3 ⫹ x
2
K(1 ⫹ h) ⫺ K(1)
.
48. If K(r) ⫽ 7 ⫺ 4r, find
h
49. If g(x) ⫽ 2 ⫺ x2, find
g(3 ⫹ h) ⫺ g(3)
.
h
P(2 ⫹ h) ⫺ P(2)
.
50. If P(m) ⫽ 2m ⫹ 3, find
h
2
51. If L(w) ⫽ ⫺2w2 ⫹ 3w ⫺ 1, find
L(⫺2 ⫹ h) ⫺ L(⫺2)
.
h
52. If D(p) ⫽ ⫺3p2 ⫺ 4p ⫹ 9, find
D(⫺1 ⫹ h) ⫺ D(⫺1)
.
h
The verbal statement “function f multiplies the square root of
the domain element by 2 and then subtracts 5 ⬙ and the
algebraic statement f (x) ⫽ 2兹x ⫺ 5 define the same
function. In Problems 53–56, translate each verbal definition
of the function into an algebraic definition.
53. Function g multiplies the domain element by 3 and then
adds 1.
f (x) ⫺ f (a)
x⫺a
In Problems 73 and 74, x ⫽ 1 is not in the domain of the
function f because the algebraic expression used to define f
does not exist at x ⫽ 1. If you were to assign a numerical value
to f at x ⫽ 1, what value would you choose? Support your
choice with information obtained by exploring the graph of f
near x ⫽ 1, by examining the numerical values of f near x ⫽ 1,
and by algebraically simplifying the expression used to define f.
73. f (x) ⫽
x2 ⫺ 1
x⫺1
74. f (x) ⫽
x3 ⫺ 1
x⫺1
APPLICATIONS
—Rate. The distance in feet that an object falls in
75. Physics—
a vacuum is given by s(t) ⫽ 16t2, where t is time in seconds. Find
(A) s(0), s(1), s(2), s(3)
s(2 ⫹ h) ⫺ s(2)
h
54. Function f multiplies the domain element by 7 and then
adds the product of 5 and the cube of the domain element.
(B)
55. Function F divides the domain element by the sum of 8
and the square root of the domain element.
(C) What happens in part (B) when h tends to 0? Interpret
physically.
56. Function G takes the square root of the sum of 4 and the
square of the domain element.
In Problems 57–60, translate each algebraic definition of the
function into a verbal definition.
57. f(x) ⫽ 2x ⫺ 3x
58. g(x) ⫽ 5x ⫺ 8x
59. F(x) ⫽ 兹x4 ⫹ 9
60. G(x) ⫽
2
—Rate. An automobile starts from rest and trav76. Physics—
els along a straight and level road. The distance in feet
traveled by the automobile is given by s(t) ⫽ 10t2, where t
is time in seconds. Find
(A) s(8), s(9), s(10), s(11)
3
(B)
x
3x ⫺ 6
C
61. Find f(x), given that f(x ⫹ h) ⫽ 2(x ⫹ h)2 ⫺ 4(x ⫹ h) ⫹ 6
62. Find g(x), given that g(x ⫹ h) ⫽ 5 ⫺ 7(x ⫹ h)2 ⫹ 8(x ⫹ h)
(C) What happens in part B as h tends to 0? Interpret
physically.
77. Boiling Point of Water. At sea level, water boils when it
reaches a temperature of 212°F. At higher altitudes, the
atmospheric pressure is lower and so is the temperature at
which water boils. The boiling point B(x) in degrees
Fahrenheit at an altitude of x feet is given approximately by
B(x) ⫽ 212 ⫺ 0.0018x
63. Find m(x), given that
(A) Complete the following table.
m(x ⫹ h) ⫽ 4(x ⫹ h) ⫺ 3兹x ⫹ h ⫹ 9
64. Find s(x), given that
s(x ⫹ h) ⫽ 2兹x ⫹ h ⫺ 6(x ⫹ h) ⫺ 5
3
s(11 ⫹ h) ⫺ s(11)
h
x
B(x)
0
5,000
10,000
15,000
20,000
25,000
30,000
43
1-3 Functions
(B) Based on the information in the table, write a brief
verbal description of the relationship between altitude
and the boiling point of water.
78. Air Temperature. As dry air moves upward, it expands
and cools. The air temperature A(x) in degrees Celsius at
an altitude of x kilometers is given approximately by
A(x) ⫽ 25 ⫺ 9x
0
1
2
3
4
t
0
1
2
3
4
Sales
5.9
6.5
7.7
8.6
9.7
S(t)
(B) Sketch by hand the graph of S and the sales data on
the same axes.
(A) Complete the following table.
x
(A) Complete the following table. Round values of S(t) to
one decimal place.
(C) Use the modeling function S to estimate the sales in
1993. In 2000.
5
A(x)
(B) Based on the information in the table, write a brief
verbal description of the relationship between altitude
and air temperature.
79. Car Rental. A car rental agency computes daily rental
charges for compact cars with the function
D(x) ⫽ 20 ⫹ 0.25x
where D(x) is the daily charge in dollars and x is the daily
mileage. Translate this algebraic statement into a verbal
statement that can be used to explain the daily charges to a
customer.
80. Installation Charges. A telephone store computes charges
for phone installation with the function
S(x) ⫽ 15 ⫹ 0.7x
where S(x) is the installation charge in dollars and x is
the time in minutes spent performing the installation.
Translate this algebraic statement into a verbal statement
that can be used to explain the installation charges to a
customer.
Merck & Co., Inc. is the world’s largest pharmaceutical
company. Problems 81–84 refer to the data in Table 7 taken
from the company’s 1993 annual report.
(D) Write a brief verbal description of the company’s
sales from 1988 to 1992.
82. Income Analysis. A mathematical model for Merck’s income is given by
I(t) ⫽ 1.2 ⫹ 0.3t
where t ⫽ 0 corresponds to 1988.
(A) Complete the following table. Round values of I(t) to
one decimal place.
t
0
1
2
3
4
Net income
1.2
1.5
1.8
2.1
2.4
I(t)
(B) Sketch by hand the graph of I and the income data on
the same axes.
(C) Use the modeling function I to estimate the income in
1993. In 2000.
(D) Write a brief verbal description of the company’s
income from 1988 to 1992.
83. Sales Analysis. A mathematical model for Merck’s sales
as a function of R & D (research & development) expenses is given by
S(r) ⫽ 0.2 ⫹ 8.6r
where r represents R & D expenditures.
T A B L E
Sales
R & D expenses
Net income
7 Selected
Financial Data
for Merck & Co., Inc.
($ in billions)
(A) Complete the following table. Round values of S(r) to
one decimal place.
r (R & D)
0.66
0.75
0.85
0.99
1.1
5.9
6.5
7.7
8.6
9.7
1988
1989
1990 1991 1992
Sales
$5.9
$6.5
$7.7
S(r)
$0.66
$0.75
$0.85 $0.99 $1.1
$1.2
$1.5
$1.8
$8.6
$2.1
$9.7
$2.4
81. Sales Analysis. A mathematical model for Merck’s sales is
given by
S(t) ⫽ 5.74 ⫹ 0.97t
where t ⫽ 0 corresponds to 1988.
(B) Sketch by hand the graph of S and the data on the
same axes.
(C) Use the modeling function S to estimate the sales if
the company spends $1.5 billion on research and
development. $2 billion.
84. Income Analysis. A mathematical model for Merck’s income as a function of R & D (research & development)
expenses is given by
44
1 FUNCTIONS AND GRAPHS
I(r) ⫽ ⫺0.5 ⫹ 2.7r
(B) Sketch by hand the graph of I and the data on the
same axes.
where r represents R & D expenditures.
(A) Complete the following table. Round values of I(r) to
one decimal place.
r (R & D)
0.66
0.75
0.85
0.99
1.1
Net income
1.2
1.5
1.8
2.1
2.4
(C) Use the modeling function I to estimate the income if
the company spends $1.5 billion on research and
development. $2 billion.
I(r)
Section 1-4 Functions: Graphs and Properties
Basic Concepts
Increasing and Decreasing Functions
Local Maxima and Minima
Piecewise-Defined Functions
The Greatest Integer Function
One of the primary goals of this course is to provide you with a set of mathematical tools that can be used, in conjunction with a graphing utility, to analyze
graphs that arise quite naturally in important applications. In this section, we discuss some basic concepts that are commonly used to describe graphs of functions.
Basic Concepts
Each function that has a real number domain and range has a graph—the graph
of the ordered pairs of real numbers that constitute the function. When functions
are graphed, domain values usually are associated with the horizontal axis and
range values with the vertical axis. Thus, the graph of a function f is the same
as the graph of the equation
y ⫽ f(x)
FIGURE 1
Graph of a function.
y or f (x)
y intercept
(x, y) or
(x, f (x))
f
y or f (x)
x
x intercept
EXAMPLE
1
where x is the independent variable and the abscissa of a point on the graph of
f. The variables y and f(x) are dependent variables, and either is the ordinate of
a point on the graph of f (see Fig. 1).
The abscissa of a point where the graph of a function intersects the x axis is
called an x intercept or zero of the function. The x intercept is also a real solution or root of the equation f(x) ⫽ 0. The ordinate of a point where the graph of
a function crosses the y axis is called the y intercept of the function. The y intercept is given by f(0), provided 0 is in the domain of f. Note that a function can
have more than one x intercept but can never have more than one y intercept—
a consequence of the vertical line test discussed in the preceding section.
Finding x and y Intercepts
Find the x and y intercepts (correct to one decimal place) of f(x) ⫽ x3 ⫹ x ⫺ 3.