Coburn J.W., Herdlick J.D. College Algebra: Graphs and Models
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3.5 The Algebra of Functions
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 func-
tions 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 ana-
lyzed. 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,
and .
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:
Domain of result
EXAMPLE 1A
䊳
Evaluating a Difference of Functions
Given ,
a. Determine the domain of . b. Find h(3) using the definition.
Solution
䊳
a. Since the domain of both f and g is , their intersection is , so the domain of
h is also .
b.
given difference
by definition
substitute 3 for
x
evaluate
multiply
subtract
result
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
䊳
For the functions f, g, and h, as defined in Example 1A,
a. Find a new function rule for h. b. Use the result to find h(3).
Solution
䊳
a. given difference
by definition
replace
f
(
x
) with and
g
(
x
) with
distribute and combine like terms x
2
7x 9
12x 921x
2
5x2 1x
2
5x2 12x 92
f
1x2 g1x2
h1x2 1f g21x2
3
6 334
39 154 36 94
3132
2
51324 32132 94
h132 f
132 g132
f
1x2 g1x2
h1x2 1f g21x2
⺢
⺢⺢
h1x2 1f g21x2
f
1x2 x
2
5x and g1x2 2x 9
P 傽 Q1f g21x2 f
1x2 g1x2
P 傽 Q1f g21x2 f
1x2 g1x2
f 1x2 g1x2 1f g21x2
f
1x2 g1x2 1f g21x2
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
340 3–60
College Algebra G&M—
cob19545_ch03_340-352.qxd 8/10/10 6:25 PM Page 340
b. 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
䊳
CAUTION
䊳
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 dif-
ference. 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.
EXAMPLE 2
䊳
Evaluating a Sum of Functions
For ,
a. Determine the domain of .
b. Find a new function rule for h.
c. Evaluate h(3).
d. Evaluate .
Solution
䊳
a. The domain of f is , while the domain of g is . Since their
intersection is , this is the domain of the new function h.
b.
given sum
by definition
substitute
x
2
for
f
(
x
) and for
g
(
x
) (no other simplifications possible)
c. substitute 3 for
x
result
d. is outside the domain of h.
Now try Exercises 11 through 14
䊳
This “intersection of domains” is illustrated in Figure 3.64.
x 1
10
h132 132
2
13 2
1x 2 x
2
1x 2
f 1x2 g1x2
h1x2 1
f g21x2
32, q2
x 僆 32, q2⺢
h112
h1x2 1
f g21x2
f
1x2 x
2
and g1x2 1x 2
3
9 21 9
h132 132
2
7132 9
3–61 Section 3.5 The Algebra of Functions 341
College Algebra G&M—
WORTHY OF NOTE
If we did try to evaluate , the
result would be , which is
not a real number. While it’s true we
could write as
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.
1 i131 13
1 13
h112
A. You’ve just seen how
we can compute a sum or
difference of functions and
determine the domain of the
result
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,
and As you might expect, for quotients we must
stipulate .g1x2 0
a
f
g
b1x2
f
1x2
g1x2
.1f
#
g21x2 f 1x2
#
g1x2
[
[
67891051112343210
67891051112343210
678910511
Intersection
12343210
Domain of g: x [2, ⬁)
Domain of h f g: x [2, ⬁)
Domain of f: x R
Figure 3.64
cob19545_ch03_340-352.qxd 8/10/10 6:25 PM Page 341
Products and Quotients of Functions
For functions f and g with domains P and Q respectively,
the product and quotient of f and g are defined by:
Domain of result
, for all
EXAMPLE 3
䊳
Computing a Product of Functions
Given and ,
a. Determine the domain of .
b. Find a new function rule for h.
c. Use the result from part (b) to evaluate h(2) and h(4).
Solution
䊳
a. The domain of f is and the domain of g is . The
intersection of these domains gives , which is the domain for h.
b.
given product
by definition
substitute for
f
and for
g
combine using properties of radicals
c. substitute 2 for
x
result
substitute 4 for
x
not a real number
The second result of Part (c) is not surprising, since is not in the domain
of h [meaning h(4) is not defined for this function].
Now try Exercises 15 through 18
䊳
In future sections, we use polynomial division as a tool for factoring, as an aid to
graphing, and to determine whether two expressions are equivalent. Understanding the
notation and domain issues related to division will strengthen our ability in these areas.
EXAMPLE 4
䊳
Computing a Quotient of Functions
Given and ,
a. Determine the domain of .
b. Find a new function rule for h.
c. Use the result from part (b) to evaluate h(0).
Solution
䊳
a. While the domain of both f and g is and their intersection is also , we know
from the definition (and past experience) that g(x) cannot be zero. The domain
of h is .
b.
given quotient
by definition
replace
f
with and
g
with x 3x
3
3x
2
2x 6
x
3
3x
2
2x 6
x 3
f
1x2
g1x2
h1x2 a
f
g
b1x2
x 僆 1q, 32 ´ 13, q2
⺢⺢
h1x2 a
f
g
b1x2
g1x2 x 3f
1x2 x
3
3x
2
2x 6
x 4
15
h142 23 2142 142
2
13 ⬇ 1.732
h122 23 2122 122
2
23 2x x
2
13 x11 x 11 x
#
13 x
f 1x2
#
g1x2
h1x2 1f
#
g21x2
x 僆 31, 34
x 僆 1q, 34x 僆 31, q2
h1x2 1f
#
g21x2
g1x2 13 x
f 1x2 11 x
g1x2 0P 傽 Q a
f
g
b1x2
f
1x2
g1x2
P 傽 Q 1f
#
g21x2 f 1x2
#
g1x2
342 CHAPTER 3 Quadratic Functions and Operations on Functions 3–62
College Algebra G&M—
cob19545_ch03_340-352.qxd 8/10/10 6:26 PM Page 342
From our work with rational expressions in Section R.5, the expression that
defines h can be simplified:
. But from the original expression, h is not defined if
, even if the result for h is a polynomial. In this case, we write the
simplified form as .
c. For we have:
replace
x
with 0
Now try Exercises 19 through 34
䊳
For additional practice with the algebra of functions, see Exercises 35 through 46.
C. Graphical and Numerical Views of Operations on Functions
The algebra of functions also has an instructive graphical interpretation, in which val-
ues for f(k) and g(k) are read from a graph (k is a given constant), with operations like
then computed and lodged (see Exercise 84).
EXAMPLE 5
䊳
Interpreting Operations on Functions Graphically
Use the graph given to find the value
of each expression:
a.
b.
c.
d.
Solution
䊳
Since the needed input values for this example are ,
4, 6, and 8, we begin by reading the value of f(x) and g(x) at
each point. From the graph, we note that and
. The other values are likewise found and appear
in the table. For we have:
a.
definition
substitute 2 for and 3 for
result
With some practice, the computations can be done mentally and we have
b.
c.
d.
Now try Exercises 47 through 56
䊳
513215
1f
#
g2142 f 142
#
g142
6
3
2
a
g
f
b 182
g182
f 182
4 112 5
1g f2162 g162 f
162
5
g 122f 122 2 3
1f g2122 f
122 g122
1
f g2122
g122 3
f
122 2
x 2
1f
#
g2142
a
g
f
b182
1g f 2162
1
f g2122
1
f g21k2 f 1k2 g1k2
h102 2
h102 102
2
2
h102
h1x2 x
2
2, x 3
g1x2 3
1x
2
221x 32
x 3
x
2
2
x
3
3x
2
2x 6
x 3
x
2
1x 32 21x 32
x 3
3–63 Section 3.5 The Algebra of Functions 343
College Algebra G&M—
B. You’ve just seen how
we can compute a product or
quotient of functions and
determine the domain
1052
4
6
x
y
(2, 3)
(2, 2)
f(x)
g(x)
xf(x) g(x)
23
45
64
863
1
3
2
cob19545_ch03_340-352.qxd 8/10/10 6:26 PM Page 343
In addition to graphically viewing operations on functions at specific points as in
Example 5, it is instructive to better understand what happens when functions are com-
bined for all possible x-values. This will shed light on certain transformations of func-
tions, as well as on the domain of the new function. For instance, consider the functions
and . The graph of f is a parabola opening downward with vertex
(0, 0), while g is a horizontal line through . For , the notation
literally says the new function is the result of adding the outputs of f and g for all val-
ues of x in the intersection of their domains. Using , and
, the result shown numerically in Table 3.2 is found by simply adding
to the output values in . The identical combination is shown graphically in
Figure 3.65, where the outputs from Y
1
are all increased by 4. Note the result h is simply
the graph f shifted up 4 units (a vertical shift).
Y
1
Y
2
4
Y
3
Y
1
Y
2
Y
1
x
2
, Y
2
4
h1x2 f
1x2 g1x2y 4
g1x2 4f
1x2 x
2
344 CHAPTER 3 Quadratic Functions and Operations on Functions 3–64
College Algebra G&M—
Table 3.2
Figure 3.65
10
10
5
4
Table 3.3
Figure 3.66
4.9
5
5
4.9
Figure 3.67
The graphical rendering also helps verify that since the domain of f and g is all real
numbers, their intersection (and the domain of h) is all real numbers.
If the same analysis is performed on functions where one or both domains are re-
stricted, the reason for the “intersection of domains” becomes clear. Using
and , we note the domain of f is
while the domain of g is . For , we have
with the result shown numerically Table 3.3 and graphically in Fig-
ures 3.66 and 3.67 (Y
3
is the bold graph). The TABLE feature of a graphing calculator
can actually be used for all three functions by activating the functions Y
1
, Y
2
, and Y
3
,
then using the right arrow to bring the column for Y
3
into view.
Y
3
Y
1
Y
2
h1x2 f 1x2 g1x2x 僆 ⺢x 僆 32, q2
g1x2 Y
2
x 1f 1x2 Y
1
21x 2
The fact that the domain of h is is clearly seen from the table. Since Y
1
is
not a real number for values less than , the combination cannot produce
real values unless . This is shown graphically in Figure 3.67, where we note the
initial point of Y
3
(in bold) is the point . Note that is also the inter-
section of the domains for f and g.
32, q212, 32
x 2
Y
1
Y
2
2
32, q2
5
6
5
5
cob19545_ch03_340-352.qxd 8/10/10 6:26 PM Page 344
EXAMPLE 6
䊳
In Example 3, we found that the product of the functions and
was . The domain of f
was stated as , the domain of g as , and the domain of h (the
intersection of domains) as . Verify these domains using a table and
graph, as in Table 3.3 and Figure 3.67.
Solution
䊳
Enter as Y
1
, as Y
2
, and Y
1
Y
2
as Y
3
. The results shown in Table 3.4
(a manual table was used to show all three functions simultaneously) and the graphs
in Figure 3.68 illustrate the domains of f and g stated, and show why the domain of
h must be their intersection. Figure 3.69 shows
the graph of Y
3
(in bold) on the same screen and
indicates the domain of h is .31, 34
13 x
11 x
x 僆 31, 34
3q, 3431, q4
h1x2 111 x213 x2
13 2x x
2
g1x2 13 x
f 1x2 11 x
3–65 Section 3.5 The Algebra of Functions 345
College Algebra G&M—
x Y
1
Y
2
Y
3
ERR: 2.450 ERR:
ERR: 2.236 ERR:
020
0 1 1.732 1.732
1 1.414 1.414 2
2 1.732 1 1.732
3200
4 2.236 ERR: ERR:
5 2.450 ERR: ERR:
1
2
3
Table 3.4
Figure 3.68
Figure 3.69
Now try Exercises 57 through 68
䊳
D. Applications of the Algebra of Functions
The algebra of functions plays an important role in the business world. For example,
the cost to manufacture an item, the revenue a company brings in, and the profit a com-
pany earns are all functions of the number of items made and sold. Further, we know a
company “breaks even” (making $0 profit) when the difference between their revenue
R and their cost C, is zero.
EXAMPLE 7
䊳
Applying Operations on Functions in Context
The fixed costs to publish Relativity Made Simple (by N.O. Way) is $2500, and the
variable cost is $4.50 per book. Marketing studies indicate the best selling price for
the book is $9.50 per copy.
a. Find the cost, revenue, and profit functions for this book.
b. Determine how many copies must be sold for the company to break even.
Solution
䊳
a. Let x represent the number of books published and sold. The cost of publishing
is $4.50 per copy, plus fixed costs (labor, storage, etc.) of $2500. The cost
function is . If the company charges $9.50 per book, the
revenue function will be . Since profit equals revenue minus costs,
substitute 9.50
x
for
R
and for
C
distribute
result
The profit function is .P1x2 5x 2500
5x 2500
9.50x 4.50x 2500
4.50x 2500 9.50x 14.50x 25002
P1x2 R1x2 C1x2
R1x2 9.50x
C1x2 4.50x 2500
WORTHY OF NOTE
In Section 1.1 we observed that the
graph of was a
semicircle with radius and
center (0, 0). It’s interesting to note
that by completing the square on
the radicand of function h in
Example 6, we can write it as
, which is a
semicircle of radius , with the
center shifted 1 unit right to
(1, 0) as shown.
r 2
y 24 1x 12
2
r a
y 2a
2
x
2
C. You’ve just seen how
we can interpret operations on
functions graphically and
numerically
cob19545_ch03_340-352.qxd 8/10/10 6:26 PM Page 345
b. When a company “breaks even,” the profit is zero: .
profit function
substitute 0 for
P
(
x
)
add 2500
divide by 5
In order for the company to break even, 500 copies must be sold.
Now try Exercises 71 through 74
䊳
There are a number of additional applications in the exercise set. See Exercises 75
through 80.
500 x
2500 5x
0 5x 2500
P1x2 5x 2500
P1x2 0
346 CHAPTER 3 Quadratic Functions and Operations on Functions 3–66
College Algebra G&M—
D. You’ve just seen how
we can apply the algebra of
functions in context
4. If , and , then
and .
5. For and
discuss/explain why the domain of
must exclude even though the resulting
quotient is the polynomial
6. Discuss/Explain the domain of ,
given and .g1x2 12 xf 1x2 1x 3
h1x2 1f
#
g21x2
2x
2
10x.
x 5,
h1x2 a
f
g
b1x2
g1x2 x 5,f
1x2 2x
3
50x
1f g h21221f g h2122
f
122 9h122 5, g122 7
䊳
CONCEPTS AND VOCABULARY
Fill in each blank with the appropriate word or phrase. Carefully reread the section, if necessary.
3.5 EXERCISES
1. Given function f with domain A and function g
with domain B, the sum can also be
written . The domain of the result
is .
2. For the product h(5) can be
found by evaluating f and g then multiplying the
result, or multiplying and evaluating the result.
Notationally these are written
and .
3. When combining functions f and g using basic
operations, the domain of the result is the
of the domains of f and g. For division,
we further stipulate that cannot equal
zero.
f
#
g
h1x2 f
1x2
#
g1x2,
f 1x2 g1x2
䊳
DEVELOPING YOUR SKILLS
7. Given and ,
(a) determine the domain for and
(b) find using the definition.
8. Given and ,
(a) determine the domain for
and (b) find h(5) using the definition.
9. For the functions f, g, and h, as defined in Exercise 7,
(a) find a new function rule for h, and (b) use the
result to find . (c) How does the result
compare to that of Exercise 7?
h122
h1x2 f
1x2 g1x2
g1x23x 7f
1x2 2x
2
18
h122
h1x2 f
1x2 g1x2
g1x2 x
2
5xf 1x2 2x
2
x 3
10. For the functions f, g, and h as defined in Exercise 8,
(a) find a new function rule for h, and (b) use the
result to find h(5). (c) How does the result compare
to that in Exercise 8?
11. For and ,
(a) determine the domain of ,
(b) find a new function rule for h, and
(c) evaluate h(4) and h(2), if possible.
12. For and ,
(a) determine the domain of ,
(b) find a new function rule for h, and
(c) evaluate h(7) and h(2), if possible.
h1x2 1 f g21x2
g1x2 12x 5
f 1x2 4x
2
2x 3
h1x2 1f g21x2
g1x2 2x
3
54f 1x2 1x 3
cob19545_ch03_340-352.qxd 11/25/10 3:40 PM Page 346
13. For and ,
(a) determine the domain of ,
(b) find a new function rule for r, and
(c) evaluate r(2) and r(4), if possible.
14. For and ,
(a) determine the domain of ,
(b) find a new function rule for r, and
(c) evaluate and r(2), if possible.
15. For and ,
(a) determine the domain of ,
(b) find a new function rule for h, and
(c) evaluate and h(21), if possible.
16. For and ,
(a) determine the domain of ,
(b) find a new function rule for h, and
(c) evaluate and h(11), if possible.
17. For and ,
(a) determine the domain of ,
(b) find a new function rule for r, and
(c) evaluate r(15) and r(3), if possible.
18. For and ,
(a) determine the domain of ,
(b) find a new function rule for r, and
(c) evaluate and , if possible.
For the functions f and g given, (a) determine the
domain of and (b) find a new function
rule for h in simplified form (if possible), noting the
domain restrictions alongside.
19. and
20. and
21. and
22. and
23. and
24. and
25. and
26. and
For the functions p and q given, (a) determine the
domain of , (b) find a new function rule for r,
and (c) use it to evaluate r(6) and , if possible.
27. and
28. and
29. and q1x2 1x 5
p1x2 x 5
q1x2 13 x
p1x2 1 x
q1x2 12 x
p1x2 2x 3
r1ⴚ62
r1x2ⴝ a
p
q
b1x2
g1x2 x 7f 1x2 x 3
g1x2 x 5f
1x2 x 1
g1x2 x 1f
1x2 x
3
1
g1x2 x 1f
1x2 x
3
7x
2
6x
g1x2 x 5f
1x2 x
3
5x
2
2x 10
g1x2 x 4f
1x2 x
3
4x
2
2x 8
g1x2 x 7f
1x2 x
2
49
g1x2 x 4f
1x2 x
2
16
h1x2ⴝ a
f
g
b1x2
r132r152
r1x2 1p
#
q21x2
q1x2 1x 4
p1x2 14 x
r1x2 1p
#
q21x2
q1x2 17 x
p1x2 1x 1
h182
h1x2 1
f
#
g21x2
g1x2 1x 7
f 1x23x 5
h142
h1x2 1f
#
g21x2
g1x2 2x 3f
1x2 1x 4
r132
r1x2 1p q21x2
q1x2 1x 2
p1x2 16 x
r1x2 1p q21x2
q1x2 13 x
p1x2 1x 5
3–67 Section 3.5 The Algebra of Functions 347
College Algebra G&M—
30. and
31. and
32. and
For the functions f and g given, (a) find a new function
rule for in simplified form. (b) If h(x)
were the original function, what would be its domain?
(c) Since we know , what
additional restrictions exist for the domain of h?
33. and
34. and
For each pair of functions f and g given, determine the
sum, difference, product, and quotient of f and g, and
determine the domain in each case.
35. and
36. and
37. and
38. and
39. and
40. and
41. and
42. and
43. and
44. and
45. and
46. and
47. Reading a graph—Used vehicle sales: The graph
given shows the number of cars C(t) and trucks T(t)
sold by Ullery Used Autos for the years 2000 to
2010. Use the graph to
estimate the number of
a. cars sold in 2005: C(5)
b. trucks sold in 2008:
T(8)
c. vehicles sold in 2009:
d. In function notation, how would you determine
how many more cars than trucks were sold in
2009? What was the actual number?
C192 T192
g1x2
1
x 5
f 1x2
4
x 3
g1x2
5
x 2
f 1x2
2
x 3
g1x2 1x 5
f 1x2 x
2
2
g1x2 1x 1
f 1x2 2x
2
g1x2 1x 6f 1x2 x 2
g1x2 1x 3
f 1x2 3x 1
g1x2 x 3f
1x2 x
2
2x 15
g1x2 x 1f
1x2 x
2
2x 3
g1x2 x 4f
1x2 x
2
3x
g1x2 3x 2f
1x2 x
2
7
g1x2 2x 3f
1x2 x 5
g1x2 x 2f
1x2 2x 3
g1x2
2x
x 2
f 1x2
4x
x 1
g1x2
3x
x 2
f 1x2
6x
x 3
h1x2ⴝ a
f
g
b1x2ⴝ
f 1x2
g1x2
h1x2ⴝ a
f
g
b1x2
q1x2 17 3xp1x2 x
2
6x
q1x2 12x 13
p1x2 x
2
36
q1x2 1x 3
p1x2 x 2
10 2345678910
1
2
3
4
5
6
7
Years since 2000
Ullery Used Auto Sales
Cars
C(t)
Number 1000
Trucks
T(t)
Exercise 47
cob19545_ch03_340-352.qxd 8/10/10 6:27 PM Page 347
51. Reading a graph—
Operations on functions:
Use the given graph to
find the result of the
operations indicated.
Note
, and so on.
a. b.
c. d.
e. f.
g. h.
i. j.
52. Reading a graph—
Operations on functions:
Use the given graph to find
the result of the operations
indicated.
Note ,
and so on.
a. b.
c. d.
e. f.
g. h.
i. j.
Some advanced applications require that we use the
algebra of functions to find a function rule for the
vertical distance between two graphs. For
and (two horizontal lines), we “see” this
vertical distance is 5 units, or in function form:
. However,
also serves as a general formula for
the vertical distance between two curves (even those that
are not horizontal lines), so long as in a
chosen interval. Find a function rule in simplified form,
for the vertical distance h(x) between the graphs of
f and g shown, for the interval indicated.
53. 54.
52
2
5
x
y
h(x)
g(x) 3
(1, 3)
f(x) 2 x 1
52
2
5
x
y
h(x)
f(x) 5
(6, 5)
g(x) x 1
2
3
x 僆 31, 74x 僆 30, 64
f 1x27 g1x2
d1x2ⴝ f 1x2ⴚ g1x2
d1x2ⴝ f 1x2ⴚ g1x2ⴝ 3 ⴚ 1ⴚ22ⴝ 5 units
g1x2ⴝⴚ2
f 1x2ⴝ 3
a
p
q
b1621p q2172
1p q21121q
#
p2122
1p
#
q2122a
p
q
b152
1p q21021p q2142
1p
#
q21121p q2142
p1122, q152 6
a
f
g
b1721f g2182
1f g21121g
#
f 2122
1f
#
g2122a
f
g
b122
1f g21021f g2142
1f
#
g21121f g2142
g1421
f
142 5,
48. Reading a graph—
Government investment:
The graph given shows a
government’s investment
in its military M(t) over
time, versus its investment
in public works P(t), in
millions of dollars. Use
the graph to estimate the amount of investment in
a. the military in 2002: M(2)
b. public works in 2005: P(5)
c. public works and the military in 2009:
d. In function notation, how would you determine
how much more will be invested in public
works than the military in 2010? What is the
actual number?
49. Reading a graph—Space
travel: The graph given
shows the revenue R(t)
and operating costs C(t)
of Space Travel Resources
(STR), for the years 2000
to 2010. Use the graph to
find the
a. revenue in 2002: R(2)
b. costs in 2008: C(8)
c. years STR broke even:
d. years costs exceeded revenue:
e. years STR made a profit:
f. For the year 2005, use function notation to
write the profit equation for STR. What was
their profit?
50. Reading a graph—
Corporate expenditures:
The graph given shows a
large corporation’s
investment in research and
development R(t) over
time, and the amount paid
to investors as dividends
D(t), in billions of dollars.
Use the graph to find the
a. dividend payments in 2002: D(2)
b. investment in 2006: R(6)
c. years where
d. years where
e. years where
f. Use function notation to write an equation for
the total expenditures of the corporation in
year t. What was the total for 2010?
R1t26 D1t2
R1t27 D1t2
R1t2 D1t2
R1t27 C1t2
C1t27 R1t2
R1t2 C1t2
M192 P192
348 CHAPTER 3 Quadratic Functions and Operations on Functions 3–68
College Algebra G&M—
10 2345678910
2
4
6
8
10
12
14
Military M(t)
Public
works
P(t)
Amount 10
6
Years since 2000
Government Investment
Exercise 48
Exercise 49
10 2345678910
1
2
3
4
5
6
7
Years since 2000
Space Travel Resources
R(t)
Revenue
Costs
C(t)
In billions of dollars
Exercise 50
10 2345678910
1
2
3
4
5
6
7
Years since 2000
Corporate Expenditures
R(t)
D(t)
In billions of dollars
8
4
4
6
x
y
f(x)
g(x)
Exercise 52
84
4
6
x
y
p(x)
q(x)
4
Exercise 51
cob19545_ch03_340-352.qxd 8/10/10 6:28 PM Page 348
3–69 Section 3.5 The Algebra of Functions 349
College Algebra G&M—
62. For and ,
a.
b.
Use the feature of a graphing calculator to
determine the domain of each newly constructed
function. Write answers in interval notation. All finite
endpoints of the domain intervals are integer values
between and 5.
63. For and ,
a.
b.
64. For and ,
a.
b.
65. For and ,
a.
b.
66. For and ,
a.
b.
67. For and ,
a.
b.
68. For and
,
a.
b. Y
1
Y
2
Y
1
Y
2
Y
2
210 2x 2
Y
1
25x 10 2
Y
1
Y
2
Y
1
Y
2
Y
2
26 2xY
1
2x 4
Y
1
Y
2
Y
1
Y
2
Y
2
1
3
x 1Y
1
4
Y
1
Y
2
Y
1
Y
2
Y
2
x 4Y
1
1
3
x 1
Y
1
Y
2
Y
1
Y
2
Y
2
24 xY
1
x
2
9
Y
1
Y
2
Y
1
Y
2
Y
2
2x 4Y
1
3
ⴚ5
GRAPH
Y
1
Y
2
Y
1
Y
2
Y
2
23 xY
1
22x 8
55. 56.
Use the TABLE feature of a graphing calculator to
determine the domain of each newly constructed
function. Write answers in interval notation. All finite
endpoints of the domain intervals are integer values
between and 5.
57. For and ,
a.
b.
58. For and ,
a.
b.
59. For and ,
a.
b.
60. For and ,
a.
b.
61. For and ,
a.
b.
Y
1
Y
2
Y
1
Y
2
Y
2
29 3xY
1
2x 2
Y
1
Y
2
Y
1
Y
2
Y
2
2x 3Y
1
9 x
2
Y
1
Y
2
Y
1
Y
2
Y
2
2x 1Y
1
x
2
1
Y
1
Y
2
Y
1
Y
2
Y
2
x 3Y
1
3x 9
Y
1
Y
2
Y
1
Y
2
Y
2
x 2Y
1
6 3x
ⴚ5
52
4
16
12
8
x
y
h(x)
f(x) ( x 3)
2
13
g(x) ( x 2)
2
(5, 9)
(0, 4)
2
2
5
x
y
h(x)
g(x)
x
(4, 4)
f(x) 5 x x
2
5
x 僆 30, 54x 僆 30, 44
䊳
WORKING WITH FORMULAS
69. Velocity of a falling object:
When air resistance is considered, the velocity (in
ft/sec) of a object weighing w lb falling for t sec
can be approximated using the formula shown
where k is the drag coefficient determined by the
aerodynamics of the object, the density of the air,
and other factors. (a) Find the velocity formula for
a 16-lb bowling ball with a drag coefficient of
. (b) Find two functions f(t) and g(t) such
that , and evaluate V(1),V(2), andV1t2 1
f g21t2
k 0.04
V ⴝ
w
k
ⴚ
w
k
e
32kt
w
V(20). (c) Considering the calculations from Part (b),
what will be the limiting or terminal velocity of the
bowling ball?
70. Surface area of a cylinder:
If the height of a cylinder is fixed at 20 cm, the
formula becomes Write this
formula in factored form and find two functions f(r)
and g(r) such that Then find A(5)
by direct calculation and also by computing the
product of f(5) and g(5), then comment on the results.
A1r2 1f
#
g21r2.
A 40r 2r
2
.
A ⴝ 2rh ⴙ 2r
2
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