Table 6 contains the average price of admission (in dollars) to a motion picture and the total box office gross (in millions of dollars) for all theaters in the United States.
S(x)150.7x D(x)200.25x h 1, 0.1, 0.01, 0.001.
s(11h)s(11) h
s(t)10t2,
where t represents time in years and corresponds to 1997.
(A) Compare the model and the data graphically and alge- braically.
(B) Estimate (to the nearest cent) the average price of admis- sion in 2006 and 2007.
96. REVENUE ANALYSISA mathematical model for the total box office gross is given by
where t represents time in years and corresponds to 1997.
(A) Compare the model and the data graphically and algebraically.
(B) Estimate (to three significant digits) the total box office gross in 2006 and 2007.
Merck & Co., Inc. is the world’s largest pharmaceutical company. Problems 97–100 refer to the data in Table 7 taken from the company’s 2005 annual report.
t0 G(t) 54.6t2802t6240
t0
Table 6 Selected Financial Data for the Motion Picture Industry
Year 1997 1999 2001 2003 2005
Average Price 4.59 5.08 5.66 6.03 6.41 of Admission ($)
Box Office Gross 6,360 7,450 8,410 9,490 8,990 ($ in millions)
Table 7 Selected Financial Data for Merck & Co., Inc.
($ in Billions) 1997 1999 2001 2003 2005
Sales 14.0 17.3 21.2 22.5 22.0
R & D Expenses 1.7 2.1 2.5 3.2 3.8
Net Income 4.6 5.9 7.3 6.8 4.6
97. SALES ANALYSISA mathematical model for Merck’s sales is given by
where t is time in years and corresponds to 1997.
(A) Compare the model and the data graphically and numerically.
(B) Estimate (to two significant digits) Merck’s sales in 2006 and in 2008.
(C) Write a brief verbal description of Merck’s sales from 1997 to 2005.
98. INCOME ANALYSISA mathematical model for Merck’s net income is given by
where t is time in years and corresponds to 1997.
(A) Compare the model and the data graphically and numerically.
(B) Estimate (to two significant digits) Merck’s net income in 2006 and in 2008.
(C) Write a brief verbal description of Merck’s net income from 1997 to 2005.
t0
I(t) 0.16t21.3t4.4 t0
S(t) 0.18t22.5t14
95. DATA ANALYSIS A mathematical model for the average price of admission to a motion picture is
A(t)0.23t4.6
100. RESEARCH AND DEVELOPMENT ANALYSIS A mathe- matical model for Merck’s net income as a function of R & D expenses is given by
where r represents R & D expenditures.
(A) Compare the model and the data graphically and numerically.
(B) Estimate (to two significant digits) Merck’s net income if the company spends $1.2 billion on research and development and if the company spends $4.2 billion.
I(r) 2.47r213.7r11.6 99. RESEARCH AND DEVELOPMENT ANALYSISA mathemati-
cal model for Merck’s sales as a function of research and devel- opment (R & D) expenses is given by
where r represents R & D expenditures.
(A) Compare the model and the data graphically and numerically.
(B) Estimate (to two significant digits) Merck’s sales if the company spends $1.2 billion on research and development and if the company spends $4.2 billion.
S(r) 3.54r223.3r15.5
Functions: Graphs and Properties
Z Basic Concepts
Z Identifying Increasing and Decreasing Functions Z Finding Local Maxima and Minima
Z Mathematical Modeling Z Defining Functions Piecewise
One of the key goals of this course is to provide you with a set of mathematical tools that can be used to analyze graphs. In many cases, these graphs will arise naturally from real-world situations. In fact, studying functions by analyzing their graphs is one of the biggest reasons that a graphing calculator is useful. In this section, we will dis- cuss some basic concepts that are commonly used to describe graphs of functions.
Z Basic Concepts
In the previous section, we saw that one way to describe a function is in terms of ordered pairs. Based on your earlier experience with graphing, this definition of func- tion may have reminded you of points on a graph, which are also described with an ordered pair of numbers. This simple connection between graphs and functions is the basis for the study of graphs of functions.
Every function that has a real-number domain and range has a graph, which is simply a pictorial representation of the ordered pairs of real numbers that make up the function. When functions are graphed, domain values are usually associated with the horizontal axis and range values with the vertical axis. In this regard, the graph of a function f is the same as the graph of the equation where x is the inde- pendent variable and the first coordinate, or abscissa, of a point on the graph of f.
The variables y and f (x) can both be used to represent the dependent variable, and either one is the second coordinate, or ordinate, of a point on the graph of f (Fig. 1).
yf (x),
1-3
ZFigure 1Graph of a function.
x y or f(x)
(x, y) or (x, f(x))
x intercept y intercept
y or f(x) f
To reflect this typical usage of variables x and y, we often refer to the abscissa of a point as the x coordinate, and the ordinate as the y coordinate.
An x coordinate of a point where the graph of a function intersects the x axis is called an x intercept of the function.
Since the height of the graph, and consequently the value of the function, at such a point is zero, the x intercepts are often referred to as real zeros of the function.
They are also solutions or roots of the equation
The y coordinate 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—otherwise it would fail the vertical line test discussed in the last section and consequently fail to be a function.
In the first section of this chapter, we solved equations of the form with a graphing calculator by graphing both sides of the equation and using the INTERSECT command. Most graphing calculators also have a ZERO or ROOT command that finds the x intercepts of a function directly from the graph of the function. Example 1 illustrates the use of this command.
f (x)c f (x)0.
EXAMPLE 1 Finding x and y Intercepts
Find the x and y intercepts (correct to three decimal places) of
SOLUTION
The y intercept occurs at the point where and from the graph of f in Figure 2, we see that it is We also see that there appears to be an x intercept between and We will use the ZERO command to find this intercept.
First we are asked to select a left bound (Fig. 3). This is a value of x to the left of the x intercept. Next we are asked to find a right bound (Fig. 4). This is a value of x to the right of the x intercept. If a function has more than one x intercept, you should select the left and right bounds so that there is only one intercept between the bounds.
x2.
x1
f (0) 4.
x0,
f (x)x3x4.
ZFigure 4
10 10
10
10
10 10
10
10
10 10
10
10
ZFigure 2 ZFigure 3
Finally, we are asked to select a guess. The guess must be between the bounds and should be close to the intercept (Fig. 5). Figure 6 shows that the x intercept (to three
decimal places) is 1.379.
Z Figure 6
10 10
10
10
10 10
10
10
ZFigure 5
MATCHED PROBLEM 1
Find the x and y intercepts (correct to three decimal places) of
f (x)x3x5.
ZZZEXPLORE-DISCUSS 1
Let
(A) Use the ZERO command on a graphing calculator to find the x inter- cepts of f.
(B) Find all solutions to the equation
(C) Discuss the differences between the graph of f, the x intercepts, and the solutions to the equation f (x)0.
x22x50.
f (x)x22x5.
The domain of a function is the set of all the x coordinates of points on the graph of the function and the range is the set of all the y coordinates. It is very useful to view the domain and range as subsets of the coordinate axes as in Figure 7.
Note the effective use of interval notation* in describing the domain and range of the functions in this figure. In Figure 7(a) a solid dot is used to indicate that a point is on the graph of the function and in Figure 7(b) an open dot to indicate that
*Interval notation is reviewed in Appendix B, Section B-1.
a point is not on the graph of the function. An open or solid dot at the end of a graph indicates that the graph terminates there, whereas an arrowhead indicates that the graph continues indefinitely beyond the portion shown with no significant changes of direction [see Fig. 7(b) and note that the arrowhead indicates that the domain extends infinitely far to the right, and the range extends infinitely far downward].
ZFigure 7 Domain and range.
x f(x)
[]
[ ]
d
b c a
Domain f [a, b] Range f [c, d]
x f(x)
(
(
d
a
Domain f (a, ) Range f (, d)
(a) (b)
EXAMPLE 2 Finding the Domain and Range from a Graph
(A) Find the domain and range of the function f whose graph is shown in Figure 8.
(B) Find f (1), f (3), and f (5).
x y or f(x)
5 4 3
3 4
1
yf(x)
ZFigure 8
SOLUTIONS
(A) The dot at the left end of the graph indicates that the graph terminates at that point, while the arrowhead on the right end indicates that the graph continues infi- nitely far to the right. So the x coordinates on the graph go from to The open dot at indicates that is not in the domain of f.
Domain: 3 6 x 6 or (3, ) 3
(3, 4)
. 3
x
4 3
5
y or f(x)
4
1
yf(x)
MATCHED PROBLEM 2
(A) Find the domain and range of the function f given by the graph in Figure 9.
(B) Find f (– 4), f (0), and f (2).
ZFigure 9
ZZZCAUTIONZZZ
When using interval notation to describe domain and range, make sure that you always write the least number first! You should find the domain by work- ing left to right along the x axis, and find the range by working bottom to top along the y axis.
Z Identifying Increasing and Decreasing Functions
ZZZEXPLORE-DISCUSS 2
Graph each function in the standard viewing window, then write a verbal description of the behavior exhibited by the graph as x moves from left to right.
(A) (B)
(C) f (x)5 (D) f (x)9x2
f (x)x3 f (x)2x
The least y coordinate on the graph is and there is no greatest y coordinate.
(The arrowhead tells us that the graph continues infinitely far upward.) The closed dot at indicates that is in the range of f.
or [ )
(B) The point on the graph with x coordinate 1 is so Likewise, and (5, 4) are on the graph, so f (3) 5and f (5) 4. (3, 5)
f (1) 4.
(1, 4), 5, Range: 5y 6
5 (3, 5)
5,
We will now take a look at increasing and decreasing properties of functions. Infor- mally, a function is increasing over an interval if its graph rises as the x coordinate
increases (moves from left to right) over that interval. A function is decreasing over an interval if its graph falls as the x coordinate increases over that interval. A func- tion is constant on an interval if its graph is horizontal over that interval (Fig. 10).
ZFigure 10 Increasing, decreasing, and constant functions.
g(x) 2x 2 x g(x)
5 5 5
5
x f(x)
5 5 5
5
f(x) x3
x h(x)
5 5 5
5
h(x) 2
x p(x)
5 5 5
5
p(x) x2 1 (a) Increasing on (, ) (b) Decreasing on (, )
(c) Constant on (, ) (d) Decreasing on Increasing on [0, )
(, 0]
More formally, we define increasing, decreasing, and constant functions as follows:
Z DEFINITION 1 Increasing, Decreasing, and Constant Functions Let I be an interval in the domain of function f. Then,
1. f is increasing on I and the graph of f is rising on I if
whenever in I.
2. f is decreasing on I and the graph of f is falling on I if
whenever in I.
3. f is constant on I and the graph of f is horizontal on I if whenever in a 6 b I.
f (a)f (b) a 6 b
f (a) 7 f (b) a 6 b
f (a) 6 f (b)
Refer to Figure 10(a) on page 51. As x moves from left to right, the values of g increase and the graph of g rises. In Figure 10(b), as x moves from left to right, the values of f decrease and the graph of f falls.
In Figure 10(c), the value of f doesn’t change (remains constant), and the graph stays at the same height. In Figure 10(d), moving from left to right, the graph falls as x increases from to 0, then rises from 0 to .
EXAMPLE 3 Describing a Graph
The graph of
is shown in Figure 11. Use the terms increasing, decreasing, rising, and falling to write a verbal description of this graph.
f (x)x312x4
ZFigure 12
x
25 5 5
25
f(x) (2, 20)
(2, 12)
x
25 5 5
25
f(x)
(3, 14)
(1, 18)
ZFigure 11 f(x)x312x4.
MATCHED PROBLEM 3
The graph of
is shown in Figure 12. Use the terms increasing, decreasing, rising, and falling to write a verbal description of this graph.
f (x) x33x29x13
SOLUTION
The values of f increase and the graph of f rises as x increases from to The values of f decrease and the graph of f falls as x increases from to 2. Finally, the values of f increase and the graph of f rises as x increases from 2 to .
2
2.
ZZZCAUTIONZZZ
The arrow on the left edge of the graph in Figure 11 does not indicate that the graph is “moving” downward. It simply tells us that there are no signif- icant changes in direction for x values less than The graph is increasing on that portion.
5.
*Maxima and minima are the plural forms of maximum and minimum, respectively.
Z DEFINITION 2 Local Maxima and Local Minima The function value f (c) is called a local max- imum if there is an interval (a, b) containing c such that
for all x in (a, b)
The function value f (c) is called a local minimum if there is an interval (a, b) containing c such that
for all x in (a, b)
The function value f (c) is called a local extremum if it is either a local maximum or a local minimum.
f (x)f (c) f (x)f (c)
b c
a x
f(x) f(c)
Local maximum
c x
f(x)
f(c)
Local minimum b a
Z Finding Local Maxima and Minima
The graph of from Example 3 (displayed in Figure 11) will help us to define some very important terms. Notice that at the point the func- tion changes from increasing to decreasing. This means that the function value is greater than any of the nearby values of the function. We will call such a point a local maximum. At the point the function changes from decreas- ing to increasing. This means that the function value is less than any nearby values of the function. We will call such a point a local minimum.
Local maxima and minima* play a crucial role in the study of graphs of functions.
For many graphs, they are the key points that determine the shape of the graph. We will also see that maxima and minima are very useful in application problems. When a function represents some quantity of interest (the profit made by a business, for example), finding the largest or smallest that quantity can get is usually very helpful.
The concepts of local maxima and minima are made more formal in the follow- ing definition:
f (2) 12 (2, 12),
f (2)20
(2, 20), f (x)x312x4
Since finding maximum or minimum values of functions is so important, most graphing calculators have commands that approximate local maxima and minima.
Examples 4 and 5 illustrate the use of these commands.
ZZZEXPLORE-DISCUSS 3
Plot the points and in a coor-
dinate plane. Draw a curve that satisfies each of the following conditions.
(A) Passes through A and B and is always increasing.
(B) Passes through A, B, and C with a local maximum at
(C) Passes through A, B, C, and D with a local maximum at and a local minimum at
What does this tell you about the connection between local extrema and increasing/decreasing properties of functions?
x7.
x3 x3.
D(10, 5) A(0, 0), B(3, 4), C(7, 1),
EXAMPLE 4 Finding Local Extrema
Find the domain, any local extrema, and the range of
Round answers to two decimal places.
SOLUTION
Because represents a real number only if the domain of f is First we must select a viewing window. Because the domain of f is we choose
We will construct a table of values on a graphing calculator (Figs. 13 and 14) to help select the remaining window variables. From Figure 13 we see that Ymin should be less than Figure 14 indicates that and Ymax greater than 70.081 should produce a reasonable view of the graph. Our choice for the win- dow variables is shown in Figure 15.
Xmax15 64.44.
Xmin0.
[ 0, ), [ 0, ).
x0, 1x
f (x)x2401x
ZFigure 13 ZFigure 14 ZFigure 15
The graph of f is shown in Figure 16. Notice that we adjusted Ymin to provide space at the bottom of the screen for the text that the graphing calculator will display.
Both the table in Figure 13 and the graph in Figure 16 indicate that f has a local minimum near After selecting the MINIMUM command on our graphing cal- culator, we are asked to select a left bound (Fig. 17), a right bound (Fig. 18), and a guess (Fig. 19). Note the arrowheads that mark the right and left boundaries.
x5.
120 0
80
15
ZFigure 16
120 0
80
15
ZFigure 17
120 0
80
15
120 0
80
15
ZFigure 18 ZFigure 19
The final graph (Fig. 20) shows that, to two decimal places, f has a local minimum value of at The curve in Figure 20 suggests that as x increases to the right without bound, the values of f (x) also increase without bound. The graph in Figure 21 and the table in Figure 22 support this suggestion. Thus, we conclude that there are no other local extrema and that the range of f is [64.63, ).
x4.64.
64.63
120 0
80
15
120 0
10,000
100
ZFigure 20 ZFigure 21 ZFigure 22
Summarizing our results, we have Domain of Range of
Local minimum: f (4.64) 64.63
f [64.63, ) f [ 0, )