Table Of ContentA 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 Aof a circle depends on the radius rof the circle. The rule that connects r
and Ais given by the equation A(cid:2)(cid:2)r2. With each positive number rthere is associ-
ated one value of A,and we say that Ais a functionof r.
B. The human population of the world Pdepends on the time t. The table gives estimates
Population
of the world population P(cid:2)t(cid:3)at time t,for certain years. For instance,
Year (millions)
P(cid:2)1950(cid:3)(cid:4)2,560,000,000
1900 1650
1910 1750 But for each value of the time tthere is a corresponding value of P,and we say that
1920 1860 Pis a function of t.
1930 2070
C. The cost Cof mailing a first-class letter depends on the weight wof the letter.
1940 2300
Although there is no simple formula that connects wand C,the post office has a rule
1950 2560
for determining Cwhen wis known.
1960 3040
1970 3710 D. The vertical acceleration aof 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 xin a set Aexactly one ele-
ment,called f(cid:2)x(cid:3),in a set B.
We usually consider functions for which the sets Aand Bare sets of real numbers. The
set Ais called the domainof the function. The number f(cid:2)x(cid:3)is the value of f at xand is
read “f of x.”The rangeof f is the set of all possible values of f(cid:2)x(cid:3)as xvaries through-
out the domain. A symbol that represents an arbitrary number in the domainof a function
f is called an independent variable. A symbol that represents a number in the rangeof f
is called a dependent variable. In Example A,for instance,ris the independent variable
and Ais the dependent variable.
It’s helpful to think of a function as a machine(see Figure 2). If xis in the domain of
x f ƒ the function f,then when xenters the machine,it’s accepted as an input and the machine
(input) (output) produces an output f(cid:2)x(cid:3) according to the rule of the function. Thus, we can think of the
domain as the set of all possible inputs and the range as the set of all possible outputs.
FIGURE 2
The preprogrammed functions in a calculator are good examples of a function as a
Machine diagram for a function ƒ
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(cid:4)0,then xis not in the
domain of this function; that is, xis not an acceptable input,and the calculator will indi-
cate an error. If x(cid:3)0,then an approximationto sxwill appear in the display. Thus,the
sxkey on your calculator is not quite the same as the exact mathematical function f defined
by f(cid:2)x(cid:3)(cid:2)sx.
x ƒ Another way to picture a function is by an arrow diagramas in Figure 3. Each arrow
connects an element of A to an element of B. The arrow indicates that f(cid:2)x(cid:3) is associated
a f(a)
with x,f(cid:2)a(cid:3)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 graphis the set of ordered pairs
f
A B (cid:5)(cid:2)x,f(cid:2)x(cid:3)(cid:3)(cid:6)x(cid:2)A(cid:7)
FIGURE 3
(Notice that these are input-output pairs.) In other words, the graph of f consists of all
Arrow diagram for ƒ
points (cid:2)x,y(cid:3)in the coordinate plane such that y(cid:2)f(cid:2)x(cid:3)and xis 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 (cid:2)x,y(cid:3)on the graph is y(cid:2)f(cid:2)x(cid:3),we can read
the value of f(cid:2)x(cid:3) 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
range y (cid:2) ƒ(x)
ƒ
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(cid:2)1(cid:3)and f(cid:2)5(cid:3).
(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 (cid:2)1,3(cid:3)lies on the graph of f,so the value of f at
1 is f(cid:2)1(cid:3)(cid:2)3. (In other words,the point on the graph that lies above x(cid:2)1 is 3 units
above the x-axis.)
When x(cid:2)5,the graph lies about 0.7 unit below the x-axis,so we estimate that
f(cid:2)5(cid:3)(cid:4)(cid:5)0.7.
|||| The notation for intervals is given in (b) We see that f(cid:2)x(cid:3)is defined when 0(cid:8)x(cid:8)7,so the domain of f is the closed inter-
Appendix A. val (cid:8)0,7(cid:9). Notice that f takes on all values from (cid:5)2 to 4,so the range of f is
(cid:6)
(cid:5)y (cid:5)2(cid:8)y(cid:8)4(cid:7)(cid:2)(cid:8)(cid:5)2,4(cid:9)
EXAMPLE 2 Sketch the graph and find the domain and range of each function.
(a) f(cid:2)x(cid:3)(cid:2)2x(cid:5)1 (b) t(cid:2)x(cid:3)(cid:2)x2
SOLUTION
y (a) The equation of the graph is y(cid:2)2x(cid:5)1,and we recognize this as being the equa-
tion of a line with slope 2 and y-intercept (cid:5)1. (Recall the slope-intercept form of the
equation of a line:y(cid:2)mx(cid:7)b. See Appendix B.) This enables us to sketch the graph of
y=2x-1 f in Figure 7. The expression 2x(cid:5)1is defined for all real numbers,so the domain of f
is the set of all real numbers,which we denote by (cid:2). The graph shows that the range is
0 1 x also (cid:2).
-1 2 (b) Since t(cid:2)2(cid:3)(cid:2)22(cid:2)4and t(cid:2)(cid:5)1(cid:3)(cid:2)(cid:2)(cid:5)1(cid:3)2(cid:2)1,we could plot the points (cid:2)2,4(cid:3)and
(cid:2)(cid:5)1,1(cid:3),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(cid:2)x2,which represents a parabola (see
FIGURE 7 Appendix C). The domain of tis (cid:2). The range of tconsists of all values of t(cid:2)x(cid:3),that is,
all numbers of the form x2. But x2(cid:3)0for all numbers xand any positive number yis a
(cid:6)
square. So the range of tis (cid:5)y y(cid:3)0(cid:7)(cid:2)(cid:8)0,(cid:6)(cid:3). This can also be seen from Figure 8.
y
(2, 4)
y=≈
(_1, 1) 1
0 1 x
FIGURE 8
❙❙❙❙
14 CHAPTER 1 FUNCTIONS AND MODELS
Representations of Functions
There are four possible ways to represent a function:
(cid:3)(cid:3) verbally (by a description in words)
(cid:3)(cid:3) numerically (by a table of values)
(cid:3)(cid:3) visually (by a graph)
(cid:3)(cid:3) algebraically (by an explicit formula)
If a single function can be represented in all four ways,it is often useful to go from one
representation to another to gain additional insight into the function. (In Example 2, for
instance, we started with algebraic formulas and then obtained the graphs.) But certain
functions are described more naturally by one method than by another. With this in mind,
let’s reexamine the four situations that we considered at the beginning of this section.
A. The most useful representation of the area of a circle as a function of its radius is
probably the algebraic formula A(cid:2)r(cid:3)(cid:2)(cid:2)r2,though it is possible to compile a table of
values or to sketch a graph (half a parabola). Because a circle has to have a positive
(cid:6)
radius,the domain is (cid:5)r r(cid:10)0(cid:7)(cid:2)(cid:2)0,(cid:6)(cid:3),and the range is also (cid:2)0,(cid:6)(cid:3).
B. We are given a description of the function in words:P(cid:2)t(cid:3)is the human population of
the world at time t. The table of values of world population on page 11 provides a
convenient representation of this function. If we plot these values,we get the graph
(called a scatter plot) in Figure 9. It too is a useful representation; the graph allows us
to absorb all the data at once. What about a formula? Of course,it’s impossible to
devise an explicit formula that gives the exact human population P(cid:2)t(cid:3)at any time t.
But it is possible to find an expression for a function that approximates P(cid:2)t(cid:3). In fact,
using methods explained in Section 1.5,we obtain the approximation
P(cid:2)t(cid:3)(cid:4)f(cid:2)t(cid:3)(cid:2)(cid:2)0.008079266(cid:3) (cid:9) (cid:2)1.013731(cid:3)t
and Figure 10 shows that it is a reasonably good “fit.”The function f is called a
mathematical modelfor population growth. In other words,it is a function with an
explicit formula that approximates the behavior of our given function. We will see,
however,that the ideas of calculus can be applied to a table of values; an explicit
formula is not necessary.
P P
6x10' 6x10'
1900 1920 1940 1960 1980 2000 t 1900 1920 1940 1960 1980 2000 t
FIGURE 9 FIGURE 10
❙❙❙❙
SECTION 1.1 FOUR WAYS TO REPRESENT A FUNCTION 15
The function Pis typical of the functions that arise whenever we attempt to apply
calculus to the real world. We start with a verbal description of a function. Then we
|||| A function defined by a table of values is may be able to construct a table of values of the function,perhaps from instrument
called a tabular function. readings in a scientific experiment. Even though we don’t have complete knowledge
of the values of the function,we will see throughout the book that it is still possible to
perform the operations of calculus on such a function.
w(ounces) C(cid:2)w(cid:3)(dollars) C. Again the function is described in words:C(cid:2)w(cid:3)is the cost of mailing a first-class letter
with weight w. The rule that the U.S. Postal Service used as of 2002 is as follows:
0(cid:4)w(cid:8)1 0.37 The cost is 37 cents for up to one ounce,plus 23 cents for each successive ounce up
1(cid:4)w(cid:8)2 0.60 to 11 ounces. The table of values shown in the margin is the most convenient repre-
2(cid:4)w(cid:8)3 0.83 sentation for this function,though it is possible to sketch a graph (see Example 10).
3(cid:4)w(cid:8)4 1.06
D. The graph shown in Figure 1 is the most natural representation of the vertical acceler-
4(cid:4)w(cid:8)5 1.29 ation function a(cid:2)t(cid:3). It’s true that a table of values could be compiled,and it is even
(cid:9) (cid:9)
(cid:9) (cid:9) possible to devise an approximate formula. But everything a geologist needs to
(cid:9) (cid:9) know—amplitudes and patterns—can be seen easily from the graph. (The same is true
for the patterns seen in electrocardiograms of heart patients and polygraphs for lie-
detection.) Figures 11 and 12 show the graphs of the north-south and east-west accel-
erations for the Northridge earthquake; when used in conjunction with Figure 1,they
provide a great deal of information about the earthquake.
a a
{cm/s@} {cm/s@}
400 200
200 100
5 10 15 20 25 30 t 5 10 15 20 25 30 t
(seconds) (seconds)
_200 _100
_400 _200
Calif. Dept. of Mines and Geology Calif. Dept. of Mines and Geology
FIGURE 11 North-south acceleration for the Northridge earthquake FIGURE 12 East-west acceleration for the Northridge earthquake
In the next example we sketch the graph of a function that is defined verbally.
EXAMPLE 3 When you turn on a hot-water faucet,the temperature T of the water depends
on how long the water has been running. Draw a rough graph of T as a function of the
T
time tthat has elapsed since the faucet was turned on.
SOLUTION The initial temperature of the running water is close to room temperature
because of the water that has been sitting in the pipes. When the water from the hot-
water tank starts coming out,T increases quickly. In the next phase,T is constant
0 t at the temperature of the heated water in the tank. When the tank is drained,T decreases
to the temperature of the water supply. This enables us to make the rough sketch of T as
FIGURE 13 a function of tin Figure13.
❙❙❙❙
16 CHAPTER 1 FUNCTIONS AND MODELS
A more accurate graph of the function in Example 3 could be obtained by using a ther-
mometer to measure the temperature of the water at 10-second intervals. In general, sci-
entists collect experimental data and use them to sketch the graphs of functions,as the next
example illustrates.
EXAMPLE 4 The data shown in the margin come from an experiment on the lactonization
t C(cid:2)t(cid:3)
of hydroxyvaleric acid at 25(cid:11)C. They give the concentration C(cid:2)t(cid:3)of this acid (in moles
0 0.0800 per liter) after t minutes. Use these data to draw an approximation to the graph of the
2 0.0570 concentration function. Then use this graph to estimate the concentration after 5minutes.
4 0.0408
SOLUTION We plot the five points corresponding to the data from the table in Figure 14.
6 0.0295
The curve-fitting methods of Section 1.2 could be used to choose a model and graph it.
8 0.0210
But the data points in Figure 14 look quite well behaved,so we simply draw a smooth
curve through them by hand as in Figure 15.
C(t) C(t)
0.08 0.08
0.06 0.06
0.04 0.04
0.02 0.02
0 1 2 3 4 5 6 7 8 t 0 1 2 3 4 5 6 7 8 t
FIGURE 14 FIGURE 15
Then we use the graph to estimate that the concentration after 5 minutes is
C(cid:2)5(cid:3)(cid:4)0.035mole(cid:10)liter
In the following example we start with a verbal description of a function in a physical
situation and obtain an explicit algebraic formula. The ability to do this is a useful skill in
solving calculus problems that ask for the maximum or minimum values of quantities.
EXAMPLE 5 A rectangular storage container with an open top has a volume of 10 m3. The
length of its base is twice its width. Material for the base costs $10 per square meter;
material for the sides costs $6 per square meter. Express the cost of materials as a func-
tion of the width of the base.
SOLUTION We draw a diagram as in Figure 16 and introduce notation by lettingwand2w
be the width and length of the base,respectively,and hbe the height.
The area of the base is (cid:2)2w(cid:3)w(cid:2)2w2,so the cost,in dollars,of the material for the
base is 10(cid:2)2w2(cid:3). Two of the sides have area whand the other two have area 2wh,so the
h
cost of the material for the sides is 6(cid:8)2(cid:2)wh(cid:3)(cid:7)2(cid:2)2wh(cid:3)(cid:9). The total cost is therefore
w C(cid:2)10(cid:2)2w2(cid:3)(cid:7)6(cid:8)2(cid:2)wh(cid:3)(cid:7)2(cid:2)2wh(cid:3)(cid:9)(cid:2)20w2(cid:7)36wh
2w
To express Cas a function of walone,we need to eliminate hand we do so by using the
FIGURE 16
fact that the volume is 10 m3. Thus
w(cid:2)2w(cid:3)h(cid:2)10
10 5
which gives h(cid:2) (cid:2)
2w2 w2
❙❙❙❙
SECTION 1.1 FOUR WAYS TO REPRESENT A FUNCTION 17
|||| In setting up applied functions as in Substituting this into the expression for C,we have
(cid:11) (cid:12)
Example5, it may be useful to review the
principles of problem solving as discussed on 5 180
page 80, particularly Step 1: Understand the C(cid:2)20w2(cid:7)36w w2 (cid:2)20w2(cid:7) w
Problem.
Therefore,the equation
180
C(cid:2)w(cid:3)(cid:2)20w2(cid:7) w(cid:10)0
w
expresses Cas a function of w.
EXAMPLE 6 Find the domain of each function.
1
(a) f(cid:2)x(cid:3)(cid:2)sx(cid:7)2 (b) t(cid:2)x(cid:3)(cid:2)
x2(cid:5)x
SOLUTION
|||| If a function is given by a formula and the (a) Because the square root of a negative number is not defined (as a real number),the
domain is not stated explicitly, the convention is domain of f consists of all values of xsuch that x(cid:7)2(cid:3)0.This is equivalent to
that the domain is the set of all numbers for x(cid:3)(cid:5)2,so the domain is the interval (cid:8)(cid:5)2,(cid:6)(cid:3).
which the formula makes sense and defines a
real number. (b) Since
1 1
t(cid:2)x(cid:3)(cid:2) (cid:2)
x2(cid:5)x x(cid:2)x(cid:5)1(cid:3)
and division by 0is not allowed,we see that t(cid:2)x(cid:3)is not defined when x(cid:2)0or x(cid:2)1.
Thus,the domain of tis
(cid:6)
(cid:5)x x(cid:3)0, x(cid:3)1(cid:7)
which could also be written in interval notation as
(cid:2)(cid:5)(cid:6), 0(cid:3)(cid:3)(cid:2)0, 1(cid:3)(cid:3)(cid:2)1, (cid:6)(cid:3)
The graph of a function is a curve in the xy-plane. But the question arises:Which curves
in the xy-plane are graphs of functions? This is answered by the following test.
The Vertical Line Test A curve in the xy-plane is the graph of a function of xif and
only if no vertical line intersects the curve more than once.
The reason for the truth of the Vertical Line Test can be seen in Figure 17. If each ver-
tical line x(cid:2)a intersects a curve only once, at (cid:2)a,b(cid:3), then exactly one functional value
is defined by f(cid:2)a(cid:3)(cid:2)b. But if a line x(cid:2)a intersects the curve twice, at (cid:2)a,b(cid:3) and (cid:2)a,c(cid:3),
then the curve can’t represent a function because a function can’t assign two different val-
ues to a.
y y
x=a x=a
(a, c)
(a, b)
(a, b)
0 a x 0 a x
FIGURE 17
❙❙❙❙
18 CHAPTER 1 FUNCTIONS AND MODELS
For example,the parabola x(cid:2)y2(cid:5)2shown in Figure 18(a) is not the graph of a func-
tion of xbecause,as you can see,there are vertical lines that intersect the parabola twice.
The parabola,however,does contain the graphs of twofunctions of x. Notice that the equa-
tion x(cid:2)y2(cid:5)2implies y2(cid:2)x(cid:7)2,so y(cid:2)(cid:12)sx(cid:7)2. Thus,the upper and lower halves
of the parabola are the graphs of the functions f(cid:2)x(cid:3)(cid:2)sx(cid:7)2 [from Example 6(a)] and
t(cid:2)x(cid:3)(cid:2)(cid:5)sx(cid:7)2.[See Figures 18(b) and (c).] We observe that if we reverse the roles of
x and y, then the equation x(cid:2)h(cid:2)y(cid:3)(cid:2)y2(cid:5)2 does define x as a function of y (with y as
the independent variable and xas the dependent variable) and the parabola now appears as
the graph of the function h.
y y y
_2
(_2, 0) 0 x _2 0 x 0 x
FIGURE 18 (a) x=¥-2 (b) y=œ„x+„„2„ (c) y=_œ„x+„„2„
Piecewise Defined Functions
The functions in the following four examples are defined by different formulas in different
parts of their domains.
EXAMPLE 7 A function f is defined by
(cid:13)
1(cid:5)x if x(cid:8)1
f(cid:2)x(cid:3)(cid:2)
x2 if x(cid:10)1
Evaluate f(cid:2)0(cid:3), f(cid:2)1(cid:3),and f(cid:2)2(cid:3)and sketch the graph.
SOLUTION Remember that a function is a rule. For this particular function the rule is the
following:First look at the value of the input x. If it happens that x (cid:8)1,then the value
of f(cid:2)x(cid:3)is 1(cid:5)x. On the other hand,if x(cid:10)1,then the value of f(cid:2)x(cid:3)is x2.
Since 0(cid:8)1, we have f(cid:2)0(cid:3)(cid:2)1(cid:5)0(cid:2)1.
Since 1(cid:8)1, we have f(cid:2)1(cid:3)(cid:2)1(cid:5)1(cid:2)0.
y
Since 2(cid:10)1, we have f(cid:2)2(cid:3)(cid:2)22(cid:2)4.
How do we draw the graph of f? We observe that if x(cid:8)1,then f(cid:2)x(cid:3)(cid:2)1(cid:5)x,so the
part of the graph of f that lies to the left of the vertical line x(cid:2)1must coincide with
1 the line y(cid:2)1(cid:5)x,which has slope (cid:5)1and y-intercept 1. If x(cid:10)1,then f(cid:2)x(cid:3)(cid:2)x2,so
the part of the graph of f that lies to the right of the line x(cid:2)1must coincide with the
1 x graph of y(cid:2)x2,which is a parabola. This enables us to sketch the graph in Figure l9.
The solid dot indicates that the point (cid:2)1,0(cid:3)is included on the graph; the open dot indi-
FIGURE 19 cates that the point (cid:2)1,1(cid:3)is excluded from the graph.
❙❙❙❙
SECTION 1.1 FOUR WAYS TO REPRESENT A FUNCTION 19
The next example of a piecewise defined function is the absolute value function. Recall
(cid:6) (cid:6)
that the absolute valueof a number a,denoted by a ,is the distance from ato 0on the
real number line. Distances are always positive or 0,so we have
|||| For a more extensive review of absolute (cid:6)a(cid:6)(cid:3)0 for every number a
values, see Appendix A.
For example,
(cid:6) (cid:6) (cid:6) (cid:6) (cid:6) (cid:6) (cid:6) (cid:6) (cid:6) (cid:6)
3 (cid:2)3 (cid:5)3 (cid:2)3 0 (cid:2)0 s2(cid:5)1 (cid:2)s2(cid:5)1 3(cid:5)(cid:2) (cid:2)(cid:2)(cid:5)3
In general,we have
(cid:6) (cid:6)
a (cid:2)a if a(cid:3)0
(cid:6) (cid:6)
a (cid:2)(cid:5)a if a(cid:4)0
(Remember that if ais negative,then (cid:5)ais positive.)
(cid:6) (cid:6)
EXAMPLE 8 Sketch the graph of the absolute value function f(cid:2)x(cid:3)(cid:2) x .
y
SOLUTION From the preceding discussion we know that
y=|x| (cid:13)
(cid:6) (cid:6) x if x(cid:3)0
x (cid:2)
(cid:5)x if x(cid:4)0
0 x Using the same method as in Example 7,we see that the graph of f coincides with the
line y(cid:2)xto the right of the y-axis and coincides with the liney(cid:2)(cid:5)xto the left of the
FIGURE 20 y-axis (see Figure 20).
EXAMPLE 9 Find a formula for the function f graphed in Figure 21.
y
1
0 1 x
FIGURE 21
SOLUTION The line through (cid:2)0,0(cid:3)and (cid:2)1,1(cid:3)has slope m(cid:2)1and y-intercept b(cid:2)0,so its
equation is y(cid:2)x. Thus,for the part of the graph of f that joins (cid:2)0,0(cid:3)to (cid:2)1,1(cid:3),we have
f(cid:2)x(cid:3)(cid:2)x if 0(cid:8) x(cid:8)1
|||| Point-slope form of the equation of a line: The line through (cid:2)1,1(cid:3)and (cid:2)2,0(cid:3)has slope m(cid:2)(cid:5)1,so its point-slope form is
y(cid:5)y (cid:2)m(cid:2)x(cid:5)x (cid:3)
1 1
y(cid:5)0(cid:2)(cid:2)(cid:5)1(cid:3)(cid:2)x(cid:5)2(cid:3) or y(cid:2)2(cid:5)x
See Appendix B.
So we have
f(cid:2)x(cid:3)(cid:2)2(cid:5)x if 1(cid:4)x(cid:8)2
