C. The Average Rate of Change Formula
Tải bản đầy đủ - 0trang
cob19545_ch03_329-339.qxd
8/10/10
8:26 PM
Page 333
College Algebra G&M—
3–53
333
Section 3.4 Quadratic Models; More on Rates of Change
Average Rates of Change Applied to Projectile Velocity
A projectile is any object that is thrown, shot, or cast upward, with no continuing
source of propulsion. The object’s height (in feet) after t sec is modeled by the function
h1t2 ϭ Ϫ16t2 ϩ vt ϩ k, where v is the initial velocity of the projectile, and k is the
height of the object at t ϭ 0. For instance, if a soccer ball is kicked vertically upward
from ground level (k ϭ 0) with an initial speed of 64 ft/sec, the height of the ball t sec
later is h1t2 ϭ Ϫ16t2 ϩ 64t. From Section 3.3, we recognize the graph will be a
parabola and evaluating the function for t ϭ 0 to 4 produces Table 3.1 and the graph
shown in Figure 3.62. Experience tells us the ball is traveling at a faster rate immediately after being kicked, as compared to when it nears its maximum height where it
¢height
momentarily stops, then begins its descent. In other words, the rate of change
¢time
has a larger value at any time prior to reaching its maximum height. To quantify this
we’ll compute the average rate of change between (a) t ϭ 0.5 and t ϭ 1, and compare
it to the average rates of change between, (b) t ϭ 1 and t ϭ 1.5, and (c) t ϭ 1.5 and
t ϭ 2.
Table 3.1
Time in
seconds
WORTHY OF NOTE
Keep in mind the graph of h
represents the relationship between
the soccer ball’s height in feet and
the elapsed time t. It does not
model the actual path of the ball.
EXAMPLE 4
ᮣ
Figure 3.62
Height in
feet
0
0
1
48
2
64
3
48
4
0
h(t)
80
(2, 64)
60
(3, 48)
(1, 48)
40
20
0
1
2
3
4
t
5
Average Rates of Change Applied to Projectiles
For the projectile function h1t2 ϭ Ϫ16t2 ϩ 64t, find the average rate of change for
a. t ʦ 30.5, 1 4 .
b. t ʦ 31, 1.54 .
c. t ʦ 31.5, 2.04 .
Then graph the secant lines representing these average rates of change and comment.
Solution
ᮣ
h1t2 2 Ϫ h1t1 2
¢h
ϭ
yields
¢t
t2 Ϫ t 1
h11.52 Ϫ h112
h122 Ϫ h11.52
¢h
¢h
b.
c.
ϭ
ϭ
¢t
1.5 Ϫ 1
¢t
2 Ϫ 1.5
60 Ϫ 48
64 Ϫ 60
ϭ
ϭ
0.5
0.5
ϭ 24
ϭ8
Using the given intervals in the formula
a.
h112 Ϫ h10.52
¢h
ϭ
¢t
1 Ϫ 10.52
48 Ϫ 28
ϭ
0.5
ϭ 40
For t ʦ 3 0.5, 1 4 , the average rate of change is 40
1,
meaning the height of the ball is increasing at an
average rate of 40 ft/sec. For t ʦ 31, 1.5 4 , the average
rate of change has slowed to 24
1 , and the soccer ball’s
height is increasing at only 24 ft/sec. In the interval
[1.5, 2], the average rate of change has slowed to
8 ft/sec. The secant lines representing these rates of
change are shown in the figure, where we note the line
from the first interval (in red), has a much steeper
slope than the line from the third interval (in gold).
h(t)
80
(2, 64)
60
(1.5, 60)
(1, 48)
40
(0.5, 28)
20
(4, 0)
(0, 0)
0
1
2
3
4
5
Now try Exercises 27 through 34
t
ᮣ
cob19545_ch03_329-339.qxd
11/25/10
3:36 PM
Page 334
College Algebra G&M—
334
3–54
CHAPTER 3 Quadratic Functions and Operations on Functions
The calculation for average rates of change can be applied to any function y ϭ f 1x2
and will yield valuable information—particulary in an applied context. For practice
with other functions, see Exercises 35 to 42.
You may have had the experience of riding in the external elevator of a modern
building, with a superb view of the surrounding area as you rise from the bottom floor.
For the first few floors, you note you can see much farther than from ground level. As
you ride to the higher floors, you can see still farther, but not that much farther due to
the curvature of the Earth. This is another example of a nonconstant rate of change.
EXAMPLE 5
ᮣ
Average Rates of Change Applied to Viewing Distance
The distance a person can see depends their elevation
above level ground. On a clear day, this viewing
distance can be approximated by the function
d1h2 ϭ 1.2 1h, where d(h) represents the viewing
distance (in miles) at height h (in feet) above level
ground. Find the average rate of change to the
nearest 100th, for
a. h ʦ 3 9, 16 4
b. h ʦ 3 196, 225 4
c. Graph the function along with the lines
representing the average rate of change and comment on what you notice.
Solution
ᮣ
d1h2 2 Ϫ d1h1 2
¢d
ϭ
¢h
h2 Ϫ h1
d12252 Ϫ d11962
¢d
b.
ϭ
¢h
225 Ϫ 196
18 Ϫ 16.8
ϭ
29
Ϸ 0.04
Use the points given in the formula:
d1162 Ϫ d192
¢d
ϭ
¢h
16 Ϫ 9
4.8 Ϫ 3.6
ϭ
7
Ϸ 0.17
0.17
¢d
c. For h ʦ 39, 16 4,
Ϸ
, meaning the viewing distance is increasing at an
¢h
1
average rate of 0.17 miles (about 898 feet) for each 1 ft increase in elevation.
¢d
0.04
For h ʦ 3 196, 225 4,
and the viewing distance is increasing at a rate
Ϸ
¢h
1
of only 0.04 miles (about 211 feet) for each increase of 1 ft. We’ll sketch the
graph using the points (0, 0), (9, 3.6), (16, 4.8), (196, 16.8) and (225, 18),
along with (100, 12) and (169, 15.6) to help round out the graph (Figure 3.63).
a.
Figure 3.63
Viewing distance
20
(225, 18)
(169, 15.6)
16
(196, 16.8)
12
(100, 12)
8
(16, 4.8)
(9, 3.6)
4
0
0
20
40
60
80
100
120
140
160
180
200
220
240
Height
C. You’ve just seen how
we can calculate average rates
of change using the average
rate of change formula
Note the slope of the secant line through the points (9, 3.6) and (16, 4.8), has a
much steeper slope than the line through (196, 16.8) and (225, 18).
Now try Exercises 51 through 56
ᮣ
cob19545_ch03_329-339.qxd
11/25/10
3:37 PM
Page 335
College Algebra G&M—
3–55
335
Section 3.4 Quadratic Models; More on Rates of Change
3.4 EXERCISES
ᮣ
CONCEPTS AND VOCABULARY
Fill in each blank with the appropriate word or phrase. Carefully reread the section if needed.
1. Data that indicate a gradual increase to a maximum
value, matched by a like decrease, might best be
modeled using
regression. In the case
described, the coefficient of the x2-term will be
.
2. For linear functions, the rate of change is
. For
functions, the rate of
change is not constant.
3. The
rate of change of a function can be
found by calculating the
of the line that
passes through two points on the graph of the
function. For nonlinear functions, this is called
a
.
4. The average rate of change of a function f(x)
between x1 and x2 is given by the formula
. To avoid division by
, x1
cannot equal x2.
5. Discuss/Explain the differences between the three
types of regressions we have studied so far (linear,
power, and quadratic). Include examples that
highlight the different behavior of each of these
types of models.
ᮣ
6. Given f 1x2 ϭ 2x2 Ϫ 12x, compare the average rate
¢y
of change near the vertex, with
on either side
¢x
of the vertex. Include specific intervals and values
in your discussion.
DEVELOPING YOUR SKILLS
Use a graphing calculator and the tables shown to find
(a) a quadratic regression equation that models the
data, (b) the output of the function for x ؍3.5, and
(c) the positive value of x where f(x) ϭ 39. Round to
nearest thousandths as necessary.
7.
9.
8.
x
x
y
1
Ϫ5.69
1
3
0.73
2
Use a graphing calculator and the tables shown to find
(a) a quadratic regression equation y ϭ f(x) that models
the data, (b) the value of f(x) at the vertex, and (c) the
two values of x where f(x) ϭ 25. Round to nearest
thousandths.
11.
12.
x
y
x
y
4.29
1
37.8
2
Ϫ48.7
10.72
3
12.1
3
Ϫ66.2
15.3
5
Ϫ59.3
y
5
18.75
4
37.74
5
7
48.37
5
58.33
6
28.2
7
8.5
7
113.67
8
71.9
8
98.6
x
y
x
y
8
67.53
x
y
10.
13.
14.
x
y
0
67.6
0
Ϫ7.6
1
67.8
1
Ϫ1.0
2
63.4
2
Ϫ13.9
5
29.6
10
30.4
9.9
20
50.1
3
68.5
3
Ϫ14.2
10
4
67.7
4
Ϫ17.5
15
3.3
30
56.7
Ϫ8.2
40
76.5
5
55.1
5
Ϫ5.1
20
6
51.7
6
3.2
25
Ϫ17.1
50
71.9
Ϫ7.9
60
73.6
7
47.3
7
27.2
30
8
31.4
8
42.6
35
4.6
70
55.4
6.8
80
53.2
9
27.2
9
88.4
40
10
9.9
10
105.3
45
25.7
90
19.3
50
54.8
100
5.2
cob19545_ch03_329-339.qxd
8/12/10
8:14 PM
Page 336
College Algebra G&M—
336
3–56
CHAPTER 3 Quadratic Functions and Operations on Functions
Using the graphs shown, find the average rate of change
of f and g for the intervals specified.
y
7
y
f(x)
(4, 0.9)
Ϫ6
(Ϫ5, Ϫ3.6)
(Ϫ3, Ϫ4)
(5, 13.5)
15
(6, 4.1)
5
(0, 0)
7 x
(2, Ϫ1.5)
(Ϫ1, Ϫ3.6)
Ϫ5
(4, 3.2)
Ϫ6
6 x
Ϫ5
(Ϫ5, Ϫ8.5)
31. Exercise 27.
g(x)
10
(Ϫ3, 3.9)
Graph the function and secant lines representing the
average rates of change for the exercises given. Comment
on what you notice in terms of the projectile’s velocity.
(1, Ϫ2.5)
Ϫ10
15. f 1x2 for x ʦ 3 Ϫ5, Ϫ1 4
16. f 1x2 for x ʦ 3 Ϫ1, 4 4
32. Exercise 28.
17. f 1x2 for x ʦ 3 2, 64
18. f 1x2 for x ʦ 3 Ϫ3, 2 4
19. f 1x2 for x ʦ 3Ϫ5, Ϫ34
20. f 1x2 for x ʦ 3 Ϫ3, Ϫ1 4
21. g1x2 for x ʦ 3Ϫ3, 14
22. g1x2 for x ʦ 3 0, 14
33. Exercise 29.
500
450
400
350
300
250
200
150
100
50
0
20
18
16
14
12
10
8
6
4
2
0
70
1
1.5
2
25. g1x2 for x ʦ 3 Ϫ3, 54
40
60
24. g1x2 for x ʦ 3 0, 44
50
30
26. g1x2 for x ʦ 3 0, 5 4
20
10
0
34. Exercise 30.
1
3
2
5
4
6
7
8
2.5
3
3.5
16
14
12
10
8
27. h1t2 ϭ Ϫ16t ϩ 160t, h in feet
a. [2, 5]
b. [3, 5]
c. [4, 5]
d. [5, 7]
2
28. h1t2 ϭ Ϫ16t ϩ 32t, h in feet
a. [0.25, 1]
b. [0.5, 1]
c. [0.75, 1]
d. [1, 1.5]
0.5
80
23. g1x2 for x ʦ 3 Ϫ3, 4 4
The functions shown are projectile equations where h(t)
¢h
is the height of the projectile after t sec. Calculate
¢t
over the intervals indicated. Include units of measurement
in your answer.
1 2 3 4 5 6 7 8 9 10
6
4
2
0
0.5
1
1.5
2
2
29. h1t2 ϭ Ϫ4.9t2 ϩ 34.3t ϩ 2.6, h in meters
a. [1, 6]
b. [2, 5]
c. [2.5, 3.5]
d. [3.5, 4.5]
30. h1t2 ϭ Ϫ4.9t2 ϩ 14.7t ϩ 4.1, h in meters
a. [1, 2]
b. [0.2, 2.8]
c. [0.5, 1.5]
d. [1.5, 2.5]
Graph each function in an appropriate window, then
find the average rate of change for the interval specified.
Round to hundredths as needed.
35. y ϭ x3 Ϫ 8; 32, 5 4
3
36. y ϭ 2
x ϩ 5; 3 Ϫ5, 34
37. y ϭ 2Ϳx ϩ 3Ϳ; 3 Ϫ4, 04
38. y ϭ Ϳ3x ϩ 1ͿϪ2; 3Ϫ2, 2 4
39. F ϭ 9.8m; [70, 100]
40. ϭ 0.2m; [1.3, 1.5]
41. A ϭ r2; [5, 7]
4
42. V ϭ r3; [5, 7]
3
cob19545_ch03_329-339.qxd
8/10/10
8:26 PM
Page 337
College Algebra G&M—
3–57
ᮣ
WORKING WITH FORMULAS
43. Height of a falling object: h1t2 ؍؊16t 2 ؊ v0 t ؉ h0
Neglecting air resistance, the height of an object
that is thrown straight downward with velocity v0
from a height of h0 is given by the formula shown,
where h(t) represents the height at time t. The
Earth’s longest vertical drop (on land) is the Rupal
Face on Nanga Parbat (Pakistan), which rises
15,000 ft above its base. From the top of this rock
face, a climber’s piton hammer slips from her hand
and is projected downward with an initial velocity
of 6 ft/sec. Determine the hammer’s height after
(a) t ϭ 5 sec and (b) t ϭ 7 sec. (c) Use the results
to calculate the average rate of change over this
ᮣ
337
Section 3.4 Quadratic Models; More on Rates of Change
2-sec interval. (d) Repeat parts a, b, and c for
t ϭ 10 sec and t ϭ 12 sec and comment.
44. The Difference Quotient: D1x2 ؍
f 1x ؉ h2 ؊ f 1x2
h
As we’ll see in Section 3.6, the difference quotient
is closely related to the average rate of change.
a. Given f 1x2 ϭ x2 and h ϭ 0.1, evaluate D(3)
using the formula.
b. Calculate the average rate of change of f (x)
over the interval [3, 3.1] and comment on what
you notice.
APPLICATIONS
45. Registration for 5-km
Registration
race: A local community
Day
Total
hosts a popular 5-km race
1
791
to raise money for breast
2
1688
cancer research. Due to
3
2407
certain legal restrictions,
4
3067
only the first 5000
registrants will be allowed
5
3692
to compete. The table
shows the cumulative number of registered
participants at the end of the day, for the first
5 days. (a) Use a graphing calculator to find a
quadratic regression equation that models the data.
Use this equation to estimate (b) the number of
participants after 1 week of registration, (c) the
number of days it will take for the race to fill up,
and (d) the maximum number of participants that
would have signed up had there been no limit.
Round to the nearest hundredth when necessary.
46. Concert tickets: In San
Ticket Sales
Francisco, the Javier
Week
Total
Mendoza Band has
1
17,751
scheduled a concert at
2
31,266
Candlestick Park. Once
3
45,311
the tickets go on sale, the
band is sure to sell out
4
54,986
this 70,000 person venue.
The table shows the cumulative number of tickets
sold each week, for the first 4 weeks. (a) Use a
graphing calculator to find a quadratic regression
equation that models the data. Use this equation to
(b) estimate the number of tickets sold after 5 weeks,
(c) estimate the number of weeks it will take for
the concert to sell out, and (d) estimate the number
of fans that won’t get to attend the show.
47. Guided tours: A tour
No. of
Start-up
guide for Kalaniohana
Tourists
Time (sec)
Tours noticed that for
2
206
groups of two to seven
4
115
people, the average time
6
63
it took to organize them at
9
79
the beginning of a tour
actually decreased as the
11
154
group size increased. For
13
269
groups of eight or more,
however, the logistics (and questions asked)
actually caused a significant increase in the start
time required. Using the given table and a
graphing calculator, (a) find a quadratic regression
equation that models the data. Use this equation to
(b) estimate how long it would take to get a group
of five tourists ready, (c) estimate the tour capacity
if start-up time can be no longer than 10 min, and
(d) estimate the fastest start time that could be
expected. Round to the nearest hundredth as
necessary.
48. Gardening: The
Water
No. of
production of a garden
Total (gal)
Tomatoes
can be diminished not
77
11
only by lack of water, but
132
25
also by overwatering.
198
29
Shay has kept diligent
2
records of her 100-ft
256
20
tomato garden’s weekly
315
1
production, as well as
the amount of water it received through watering
and rain. Use the given table and a graphing
calculator to (a) find a quadratic regression
equation that models the data. Then use this
equation to (b) estimate how many tomatoes she
cob19545_ch03_329-339.qxd
8/10/10
8:26 PM
Page 338
College Algebra G&M—
338
CHAPTER 3 Quadratic Functions and Operations on Functions
can expect when the garden receives 156 gal of
water per week, (c) estimate how much water
the garden received if there were 15 tomatoes
produced per week, and (d) estimate the maximum
number of tomatoes she can expect from the garden
in a week. Round to the nearest ten-thousandth as
necessary.
49. Weight of a
fetus: The
growth rate of a
Full term
fetus in the
(40, 3200)
mother’s womb
(36, 2600)
(by weight in
grams) is
(32, 1600)
modeled by the
(29, 1100)
graph shown
(25, 900)
here, beginning
with the 25th
Age (weeks)
week of
gestation. (a) Calculate the average rate of change
(slope of the secant line) between the 25th week
and the 29th week. Is the slope of the secant line
positive or negative? Discuss what the slope means
in this context. (b) Is the fetus gaining weight faster
between the 25th and 29th week, or between the
32nd and 36th week? Compare the slopes of both
secant lines and discuss.
3800
3600
3400
Weight (g)
3200
3–58
For Exercises 51 to 56, use the formula for the average
f 1x2 2 ؊ f 1x1 2
rate of change
.
x2 ؊ x1
51. Average rate of change: For f 1x2 ϭ x3, (a) calculate
the average rate of change for the interval x ϭ Ϫ2
to x ϭ Ϫ1 and (b) calculate the average rate of
change for the interval x ϭ 1 to x ϭ 2. (c) What do
you notice about the answers from parts (a) and
(b)? (d) Sketch the graph of this function along
with the lines representing these average rates of
change and comment on what you notice.
2800
2400
2000
1600
1200
800
24
26
28
30
32
34
36
38
40
42
50. Fertility rates:
Over the years,
(60, 3.6)
fertility rates for
(10, 3.4)
women in the
(20, 3.2)
(50, 3.0)
United States
(70, 2.4)
(average number
of children per
(40, 2.2)
(90, 2.0)
(80, 1.8)
woman) have
varied a great
deal, though in
the twenty-first
Year (10 → 1910)
century they’ve
begun to level out. The graph shown models this
fertility rate for most of the twentieth century.
(a) Calculate the average rate of change from the
years 1920 to 1940. Is the slope of the secant line
positive or negative? Discuss what the slope means
in this context. (b) Calculate the average rate of
change from the year 1940 to 1950. Is the slope of
the secant line positive or negative? Discuss what
the slope means in this context. (c) Was the fertility
rate increasing faster from 1940 to 1950, or from
1980 to 1990? Compare the slope of both secant
lines and comment.
Rate (children per woman)
4.0
3.0
2.0
52. Average rate of change: Knowing the general
3
shape of the graph for f 1x2 ϭ 1x, (a) is the
average rate of change greater between x ϭ 0 and
x ϭ 1 or between x ϭ 7 and x ϭ 8? Why?
(b) Calculate the rate of change for these intervals
and verify your response. (c) Approximately how
many times greater is the rate of change?
53. Height of an arrow: If an
arrow is shot vertically from a
bow with an initial speed of
192 ft/sec, the height of the
arrow can be modeled by the
function h1t2 ϭ Ϫ16t2 ϩ 192t,
where h(t) represents the height of the arrow
after t sec (assume the arrow was shot from
ground level).
a. What is the arrow’s height at t ϭ 1 sec?
b. What is the arrow’s height at t ϭ 2 sec?
c. What is the average rate of change from t ϭ 1
to t ϭ 2?
d. What is the rate of change from t ϭ 10 to
t ϭ 11? Why is it the same as (c) except for
the sign?
1.0
10
20
30
40
50
60
70
80
90
100
110
Source: Statistical History of the United States from Colonial Times to Present
54. Height of a water rocket: Although they have
been around for decades, water rockets continue to
be a popular toy. A plastic rocket is filled with
water and then pressurized using a handheld pump.
The rocket is then released and off it goes! If the
rocket has an initial velocity of 96 ft/sec, the height
of the rocket can be modeled by the function
h1t2 ϭ Ϫ16t2 ϩ 96t, where h(t) represents the
height of the rocket after t sec (assume the rocket
was shot from ground level).
a. Find the rocket’s height at t ϭ 1 and
t ϭ 2 sec.
b. Find the rocket’s height at t ϭ 3 sec.
c. Would you expect the average rate of change
to be greater between t ϭ 1 and t ϭ 2, or
between t ϭ 2 and t ϭ 3? Why?
d. Calculate each rate of change and discuss your
answer.
cob19545_ch03_329-339.qxd
8/10/10
8:26 PM
Page 339
College Algebra G&M—
3–59
55. Velocity of a falling object: The impact velocity
of an object dropped from a height is modeled by
v ϭ 12gs, where v is the velocity in feet per
second (ignoring air resistance), g is the
acceleration due to gravity (32 ft/sec2 near the
Earth’s surface), and s is the height from which
the object is dropped.
a. Find the velocity at s ϭ 5 ft and s ϭ 10 ft.
b. Find the velocity at s ϭ 15 ft and s ϭ 20 ft.
c. Would you expect the average rate of change
to be greater between s ϭ 5 and s ϭ 10, or
between s ϭ 15 and s ϭ 20?
d. Calculate each rate of change and discuss your
answer.
ᮣ
339
56. Temperature drop: One day in November, the town
of Coldwater was hit by a sudden winter storm that
caused temperatures to plummet. During the storm,
the temperature T (in degrees Fahrenheit) could be
modeled by the function T1h2 ϭ 0.8h2 Ϫ 16h ϩ 60,
where h is the number of hours since the storm
began. Graph the function and use this information
to answer the following questions.
a. What was the temperature as the storm began?
b. How many hours until the temperature dropped
below zero degrees?
c. How many hours did the temperature remain
below zero?
d. What was the coldest temperature recorded
during this storm?
EXTENDING THE CONCEPT
57. The function A1t2 ϭ 200t gives the amount of air
(in cubic inches) that a compressor has pumped out
after t sec. The volume of a spherical balloon being
4
inflated by this compressor is given by V1r2 ϭ r3,
3
with the radius of the balloon modeled by
r1t2 ϭ Ϫ0.02t2 ϩ 0.76t ϩ 2.26, where r(t) is the
radius after t sec.
a. If the balloon pops when the radius is at its
maximum, what is the maximum volume of
the balloon?
b. What amount of air (the volume) was needed
to pop the balloon?
¢r
c. Calculate the average rates of change
and
¢t
¢V
during the first second of inflation and the
¢r
last second of inflation. Compare the results.
ᮣ
Section 3.4 Quadratic Models; More on Rates of Change
58. In Exercise 44, you were provided with a formula
called the difference quotient. This formula can be
derived by finding the average rate of change of a
function f (x) over an interval 3x, x ϩ h4 .
a. Use the definition of average rate of change to
derive the formula for the difference quotient.
b. Find and simplify the difference quotient of
f 1x2 ϭ x2 ϩ 3x.
59. The floor function f 1x2 ϭ :x ; and the ceiling
function g1x2 ϭ
produce some interesting average rates of change.
The average rate of change of these two functions
over any interval 1 unit or longer must lie within
what range of values?
MAINTAINING YOUR SKILLS
60. (1.1) Complete the squares in x and y to find the
center and radius of the circle defined by
x2 ϩ y2 ϩ 6x Ϫ 8y ϭ 0. Then graph the circle on a
graphing calculator.
61. (1.5) Solve the following inequalities graphically
using the Intersection-of-Graphs method. Round to
nearest hundredths when necessary.
xϪ5
6 2
a.
xϪ2
2
b. x Ϫ 13 Ϫ x2 7 710.2x ϩ 0.52
5
62. (3.3) Find an equation of the quadratic function
with vertex 1Ϫ5, Ϫ32 and y-intercept (0, 7).
|x ϩ 3| x 6 Ϫ2
63. (2.2/2.5) Given f 1x2 ϭ • 1
, find
Ϫ x
x Ն Ϫ2
2
the equation of g(x), given its graph is the same as
f(x) but translated right 3 units and reflected across
the x-axis.
cob19545_ch03_340-352.qxd
8/10/10
6:25 PM
Page 340
College Algebra G&M—
3.5
The Algebra of Functions
LEARNING OBJECTIVES
In Section 3.5 you will see how we can:
A. Compute a sum or
difference of functions
and determine the
domain of the result
B. Compute a product or
quotient of functions and
determine the domain
C. Interpret operations on
functions graphically and
numerically
D. Apply the algebra of
functions in context
In Section 2.2, we created new functions graphically by applying transformations to
basic functions. In this section, we’ll use two (or more) functions to create new functions algebraically. Previous courses often contain material on the sum, difference,
product, and quotient of polynomials. Here we’ll combine functions with the basic
operations, noting the result is also a function that can be evaluated, graphed, and analyzed. We call these basic operations on functions the algebra of functions.
A. Sums and Differences of Functions
This section introduces the notation used for basic operations on functions. Here we’ll
note the result is also a function whose domain depends on the original functions. In
general, if f and g are functions with overlapping domains, f 1x2 ϩ g1x2 ϭ 1 f ϩ g21x2
and f 1x2 Ϫ g1x2 ϭ 1 f Ϫ g2 1x2 .
Sums and Differences of Functions
For functions f and g with domains P and Q respectively,
the sum and difference of f and g are defined by:
1 f ϩ g21x2 ϭ f 1x2 ϩ g1x2
Domain of result
1 f Ϫ g21x2 ϭ f 1x2 Ϫ g1x2
EXAMPLE 1A
Solution
ᮣ
ᮣ
PʝQ
PʝQ
Evaluating a Difference of Functions
Given f 1x2 ϭ x2 Ϫ 5x and g1x2 ϭ 2x Ϫ 9,
a. Determine the domain of h1x2 ϭ 1 f Ϫ g21x2 .
b. Find h(3) using the definition.
a. Since the domain of both f and g is ޒ, their intersection is ޒ, so the domain of
h is also ޒ.
b. h1x2 ϭ 1 f Ϫ g21x2
given difference
by definition
ϭ f 1x2 Ϫ g1x2
substitute 3 for x
h132 ϭ f 132 Ϫ g132
2
ϭ 3 132 Ϫ 5132 4 Ϫ 32132 Ϫ 9 4 evaluate
multiply
ϭ 39 Ϫ 15 4 Ϫ 36 Ϫ 94
subtract
ϭ Ϫ6 Ϫ 3 Ϫ34
result
ϭ Ϫ3
If the function h is to be graphed or evaluated numerous times, it helps to compute
a new function rule for h, rather than repeatedly apply the definition.
EXAMPLE 1B
ᮣ
For the functions f, g, and h, as defined in Example 1A,
a. Find a new function rule for h.
Solution
340
ᮣ
a. h1x2 ϭ 1 f Ϫ g21x2
ϭ f 1x2 Ϫ g1x2
ϭ 1x2 Ϫ 5x2 Ϫ 12x Ϫ 92
ϭ x2 Ϫ 7x ϩ 9
b. Use the result to find h(3).
given difference
by definition
replace f (x ) with 1x 2 Ϫ 5x2 and g(x) with 12x Ϫ 92
distribute and combine like terms
3–60
cob19545_ch03_340-352.qxd
8/10/10
6:25 PM
Page 341
College Algebra G&M—
3–61
341
Section 3.5 The Algebra of Functions
b. h132 ϭ 132 2 Ϫ 7132 ϩ 9
ϭ 9 Ϫ 21 ϩ 9
ϭ Ϫ3
substitute 3 for x
multiply
result
Notice the result from Part (b) is identical to that in Example 1A.
Now try Exercises 7 through 10 ᮣ
ᮣ
CAUTION
EXAMPLE 2
ᮣ
Solution
ᮣ
WORTHY OF NOTE
If we did try to evaluate h1Ϫ12 , the
result would be 1 ϩ 1Ϫ3, which is
not a real number. While it’s true we
could write 1 ϩ 1Ϫ3 as 1 ϩ i13
and consider it an “answer,” our
study here focuses on real numbers
and the graphs of functions in a
coordinate system where x and y
are both real.
From Example 1A, note the importance of using grouping symbols with the algebra of
functions. Without them, we could easily confuse the signs of g when computing the difference. Also, note that any operation applied to the functions f and g simply results in an
expression representing a new function rule for h, and is not an equation that needs to be
factored or solved.
Evaluating a Sum of Functions
For f 1x2 ϭ x2 and g1x2 ϭ 1x Ϫ 2,
a. Determine the domain of h1x2 ϭ 1 f ϩ g21x2 .
b. Find a new function rule for h.
c. Evaluate h(3).
d. Evaluate h1Ϫ12 .
a. The domain of f is ޒ, while the domain of g is x ʦ 32, q 2 . Since their
intersection is 3 2, q 2 , this is the domain of the new function h.
b. h1x2 ϭ 1 f ϩ g21x2
given sum
ϭ f 1x2 ϩ g1x2
by definition
ϭ x2 ϩ 1x Ϫ 2
substitute x2 for f (x ) and 1x Ϫ 2 for g (x ) (no other simplifications possible)
c. h132 ϭ 132 2 ϩ 13 Ϫ 2 substitute 3 for x
result
ϭ 10
d. x ϭ Ϫ1 is outside the domain of h.
Now try Exercises 11 through 14 ᮣ
This “intersection of domains” is illustrated in Figure 3.64.
Figure 3.64
Domain of f: x ⑀ R
Domain of g: x ⑀ [2, ϱ)
A. You’ve just seen how
we can compute a sum or
difference of functions and
determine the domain of the
result
Ϫ3 Ϫ2 Ϫ1
0
1
2
Ϫ3 Ϫ2 Ϫ1
0
1
2
3
4
5
6
7
8
9
10
11
3
4
5
6
7
8
9
10
11
9
10
11
[
Intersection
Domain of h ϭ f ϩ g: x ⑀ [2, ϱ)
Ϫ3 Ϫ2 Ϫ1
[
0
1
2
3
4
5
6
7
8
B. Products and Quotients of Functions
The product and quotient of two functions is defined in a manner similar to that for
sums and differences. For example, if f and g are functions with overlapping domains,
f 1x2
f
1 f # g21x2 ϭ f 1x2 # g1x2 and a b1x2 ϭ
. As you might expect, for quotients we must
g
g1x2
stipulate g1x2 0.