A graphical representation of a function––here the
number of hours of daylight as a function of the time
of year at various latitudes–– is often the most nat-
ural and convenient way to represent the function.
Functions and Models
, The fundamental objects that we deal with in calculus are
functions. This chapter prepares the way for calculus by
discussing the basic ideas concerning functions, their
graphs, and ways of transforming and combining them.
We stress that a function can be represented in different
ways: by an equation, in a table, by a graph, or in words. We look at the main
types of functions that occur in calculus and describe the process of using these func-
tions as mathematical models of real-world phenomena. We also discuss the use of
graphing calculators and graphing software for computers.
|||| 1.1 Four Ways to Represent a Function
Functions arise whenever one quantity depends on another. Consider the following four
situations.
A. The area A of a circle depends on the radius r of the circle. The rule that connects r
and A is given by the equation A r 2. With each positive number r there is associ-
ated one value of A, and we say that A is a function of r.
B. The human population of the world P depends on the time t. The table gives estimates
Population
Year (millions) of the world population Pt at time t, for certain years. For instance,
1900 1650
P1950 2,560,000,000
1910 1750 But for each value of the time t there is a corresponding value of P, and we say that
1920 1860 P is a function of t.
1930 2070
C. The cost C of mailing a first-class letter depends on the weight w of the letter.
1940 2300
Although there is no simple formula that connects w and C, the post office has a rule
1950 2560
1960 3040
for determining C when w is known.
1970 3710 D. The vertical acceleration a of the ground as measured by a seismograph during an
1980 4450 earthquake is a function of the elapsed time t. Figure 1 shows a graph generated by
1990 5280 seismic activity during the Northridge earthquake that shook Los Angeles in 1994.
2000 6080 For a given value of t, the graph provides a corresponding value of a.
a
{cm/s@}
100
50
5 10 15 20 25 30 t (seconds)
FIGURE 1 _50
Vertical ground acceleration during
the Northridge earthquake Calif. Dept. of Mines and Geology
,12 ❙❙❙❙ CHAPTER 1 FUNCTIONS AND MODELS
Each of these examples describes a rule whereby, given a number (r, t, w, or t), another
number ( A, P, C, or a) is assigned. In each case we say that the second number is a func-
tion of the first number.
A function f is a rule that assigns to each element x in a set A exactly one ele-
ment, called f x, in a set B.
We usually consider functions for which the sets A and B are sets of real numbers. The
set A is called the domain of the function. The number f x is the value of f at x and is
read “ f of x.” The range of f is the set of all possible values of f x as x varies through-
out the domain. A symbol that represents an arbitrary number in the domain of a function
f is called an independent variable. A symbol that represents a number in the range of f
is called a dependent variable. In Example A, for instance, r is the independent variable
and A is the dependent variable.
It’s helpful to think of a function as a machine (see Figure 2). If x is in the domain of
x f ƒ the function f, then when x enters the machine, it’s accepted as an input and the machine
(input) (output) produces an output f x according to the rule of the function. Thus, we can think of the
FIGURE 2
domain as the set of all possible inputs and the range as the set of all possible outputs.
Machine diagram for a function ƒ The preprogrammed functions in a calculator are good examples of a function as a
machine. For example, the square root key on your calculator computes such a function.
You press the key labeled s (or sx ) and enter the input x. If x 0, then x is not in the
domain of this function; that is, x is not an acceptable input, and the calculator will indi-
cate an error. If x 0, then an approximation to sx will appear in the display. Thus, the
sx key on your calculator is not quite the same as the exact mathematical function f defined
by f x sx.
x ƒ Another way to picture a function is by an arrow diagram as in Figure 3. Each arrow
a f(a) connects an element of A to an element of B. The arrow indicates that f x is associated
with x, f a is associated with a, and so on.
The most common method for visualizing a function is its graph. If f is a function with
domain A, then its graph is the set of ordered pairs
f
A B
x, f x x A
FIGURE 3
Arrow diagram for ƒ (Notice that these are input-output pairs.) In other words, the graph of f consists of all
points x, y in the coordinate plane such that y f x and x is in the domain of f .
The graph of a function f gives us a useful picture of the behavior or “life history” of
a function. Since the y-coordinate of any point x, y on the graph is y f x, we can read
the value of f x from the graph as being the height of the graph above the point x (see
Figure 4). The graph of f also allows us to picture the domain of f on the x-axis and its
range on the y-axis as in Figure 5.
y { x, ƒ} y
ƒ
range y ƒ(x)
f (2)
f (1)
0 1 2 x x 0 x
domain
FIGURE 4 FIGURE 5
, SECTION 1.1 FOUR WAYS TO REPRESENT A FUNCTION ❙❙❙❙ 13
EXAMPLE 1 The graph of a function f is shown in Figure 6.
(a) Find the values of f 1 and f 5.
(b) What are the domain and range of f ?
y
1
0 1 x
FIGURE 6
SOLUTION
(a) We see from Figure 6 that the point 1, 3 lies on the graph of f , so the value of f at
1 is f 1 3. (In other words, the point on the graph that lies above x 1 is 3 units
above the x-axis.)
When x 5, the graph lies about 0.7 unit below the x-axis, so we estimate that
f 5 0.7.
|||| The notation for intervals is given in (b) We see that f x is defined when 0 x 7, so the domain of f is the closed inter-
Appendix A. val 0, 7 . Notice that f takes on all values from 2 to 4, so the range of f is
y 2 y 4 2, 4
EXAMPLE 2 Sketch the graph and find the domain and range of each function.
(a) fx 2x 1 (b) tx x 2
SOLUTION
y (a) The equation of the graph is y 2x 1, and we recognize this as being the equa-
tion of a line with slope 2 and y-intercept 1. (Recall the slope-intercept form of the
equation of a line: y mx b. See Appendix B.) This enables us to sketch the graph of
y=2 x-1 f in Figure 7. The expression 2x 1 is defined for all real numbers, so the domain of f
is the set of all real numbers, which we denote by . The graph shows that the range is
0 1 x also .
2
-1 (b) Since t2 2 2 4 and t1 12 1, we could plot the points 2, 4 and
1, 1, together with a few other points on the graph, and join them to produce the
graph (Figure 8). The equation of the graph is y x 2, which represents a parabola (see
FIGURE 7
Appendix C). The domain of t is . The range of t consists of all values of tx, that is,
all numbers of the form x 2. But x 2 0 for all numbers x and any positive number y is a
square. So the range of t is y y 0 0, . This can also be seen from Figure 8.
y
(2, 4)
y=≈
(_1, 1) 1
0 1 x
FIGURE 8
number of hours of daylight as a function of the time
of year at various latitudes–– is often the most nat-
ural and convenient way to represent the function.
Functions and Models
, The fundamental objects that we deal with in calculus are
functions. This chapter prepares the way for calculus by
discussing the basic ideas concerning functions, their
graphs, and ways of transforming and combining them.
We stress that a function can be represented in different
ways: by an equation, in a table, by a graph, or in words. We look at the main
types of functions that occur in calculus and describe the process of using these func-
tions as mathematical models of real-world phenomena. We also discuss the use of
graphing calculators and graphing software for computers.
|||| 1.1 Four Ways to Represent a Function
Functions arise whenever one quantity depends on another. Consider the following four
situations.
A. The area A of a circle depends on the radius r of the circle. The rule that connects r
and A is given by the equation A r 2. With each positive number r there is associ-
ated one value of A, and we say that A is a function of r.
B. The human population of the world P depends on the time t. The table gives estimates
Population
Year (millions) of the world population Pt at time t, for certain years. For instance,
1900 1650
P1950 2,560,000,000
1910 1750 But for each value of the time t there is a corresponding value of P, and we say that
1920 1860 P is a function of t.
1930 2070
C. The cost C of mailing a first-class letter depends on the weight w of the letter.
1940 2300
Although there is no simple formula that connects w and C, the post office has a rule
1950 2560
1960 3040
for determining C when w is known.
1970 3710 D. The vertical acceleration a of the ground as measured by a seismograph during an
1980 4450 earthquake is a function of the elapsed time t. Figure 1 shows a graph generated by
1990 5280 seismic activity during the Northridge earthquake that shook Los Angeles in 1994.
2000 6080 For a given value of t, the graph provides a corresponding value of a.
a
{cm/s@}
100
50
5 10 15 20 25 30 t (seconds)
FIGURE 1 _50
Vertical ground acceleration during
the Northridge earthquake Calif. Dept. of Mines and Geology
,12 ❙❙❙❙ CHAPTER 1 FUNCTIONS AND MODELS
Each of these examples describes a rule whereby, given a number (r, t, w, or t), another
number ( A, P, C, or a) is assigned. In each case we say that the second number is a func-
tion of the first number.
A function f is a rule that assigns to each element x in a set A exactly one ele-
ment, called f x, in a set B.
We usually consider functions for which the sets A and B are sets of real numbers. The
set A is called the domain of the function. The number f x is the value of f at x and is
read “ f of x.” The range of f is the set of all possible values of f x as x varies through-
out the domain. A symbol that represents an arbitrary number in the domain of a function
f is called an independent variable. A symbol that represents a number in the range of f
is called a dependent variable. In Example A, for instance, r is the independent variable
and A is the dependent variable.
It’s helpful to think of a function as a machine (see Figure 2). If x is in the domain of
x f ƒ the function f, then when x enters the machine, it’s accepted as an input and the machine
(input) (output) produces an output f x according to the rule of the function. Thus, we can think of the
FIGURE 2
domain as the set of all possible inputs and the range as the set of all possible outputs.
Machine diagram for a function ƒ The preprogrammed functions in a calculator are good examples of a function as a
machine. For example, the square root key on your calculator computes such a function.
You press the key labeled s (or sx ) and enter the input x. If x 0, then x is not in the
domain of this function; that is, x is not an acceptable input, and the calculator will indi-
cate an error. If x 0, then an approximation to sx will appear in the display. Thus, the
sx key on your calculator is not quite the same as the exact mathematical function f defined
by f x sx.
x ƒ Another way to picture a function is by an arrow diagram as in Figure 3. Each arrow
a f(a) connects an element of A to an element of B. The arrow indicates that f x is associated
with x, f a is associated with a, and so on.
The most common method for visualizing a function is its graph. If f is a function with
domain A, then its graph is the set of ordered pairs
f
A B
x, f x x A
FIGURE 3
Arrow diagram for ƒ (Notice that these are input-output pairs.) In other words, the graph of f consists of all
points x, y in the coordinate plane such that y f x and x is in the domain of f .
The graph of a function f gives us a useful picture of the behavior or “life history” of
a function. Since the y-coordinate of any point x, y on the graph is y f x, we can read
the value of f x from the graph as being the height of the graph above the point x (see
Figure 4). The graph of f also allows us to picture the domain of f on the x-axis and its
range on the y-axis as in Figure 5.
y { x, ƒ} y
ƒ
range y ƒ(x)
f (2)
f (1)
0 1 2 x x 0 x
domain
FIGURE 4 FIGURE 5
, SECTION 1.1 FOUR WAYS TO REPRESENT A FUNCTION ❙❙❙❙ 13
EXAMPLE 1 The graph of a function f is shown in Figure 6.
(a) Find the values of f 1 and f 5.
(b) What are the domain and range of f ?
y
1
0 1 x
FIGURE 6
SOLUTION
(a) We see from Figure 6 that the point 1, 3 lies on the graph of f , so the value of f at
1 is f 1 3. (In other words, the point on the graph that lies above x 1 is 3 units
above the x-axis.)
When x 5, the graph lies about 0.7 unit below the x-axis, so we estimate that
f 5 0.7.
|||| The notation for intervals is given in (b) We see that f x is defined when 0 x 7, so the domain of f is the closed inter-
Appendix A. val 0, 7 . Notice that f takes on all values from 2 to 4, so the range of f is
y 2 y 4 2, 4
EXAMPLE 2 Sketch the graph and find the domain and range of each function.
(a) fx 2x 1 (b) tx x 2
SOLUTION
y (a) The equation of the graph is y 2x 1, and we recognize this as being the equa-
tion of a line with slope 2 and y-intercept 1. (Recall the slope-intercept form of the
equation of a line: y mx b. See Appendix B.) This enables us to sketch the graph of
y=2 x-1 f in Figure 7. The expression 2x 1 is defined for all real numbers, so the domain of f
is the set of all real numbers, which we denote by . The graph shows that the range is
0 1 x also .
2
-1 (b) Since t2 2 2 4 and t1 12 1, we could plot the points 2, 4 and
1, 1, together with a few other points on the graph, and join them to produce the
graph (Figure 8). The equation of the graph is y x 2, which represents a parabola (see
FIGURE 7
Appendix C). The domain of t is . The range of t consists of all values of tx, that is,
all numbers of the form x 2. But x 2 0 for all numbers x and any positive number y is a
square. So the range of t is y y 0 0, . This can also be seen from Figure 8.
y
(2, 4)
y=≈
(_1, 1) 1
0 1 x
FIGURE 8
