Download SECTION 2-5 Combining Functions

2-5 Combining Functions
★★ 91.
Physics. A stunt driver is planning to jump a motorcycle
from one ramp to another as illustrated in the figure. The
ramps are 10 feet high, and the distance between the ramps
is 80 feet. The trajectory of the cycle through the air is
given by the graph of
★★
165
92. Physics. The trajectory of a circus performer shot from a
cannon is given by the graph of the function
f (x) x 16
1
f (x) x 2 x2
4
v
1 2
x
100
Both the cannon and the net are 10 feet high (see the figure).
f (x)
where v is the velocity of the cycle in feet per second as it
leaves the ramp.
f(x)
x
10 feet
x
10 feet
80 feet
(A) How fast must the cycle be traveling when it leaves the
ramp in order to follow the trajectory illustrated in the
figure?
(B) What is the maximum height of the cycle above the
ground as it follows this trajectory?
SECTION
2-5
(A) How far from the muzzle of the cannon should the center of the net be placed so that the performer lands in
the center of the net?
(B) What is the maximum height of the performer above
the ground?
Combining Functions
•
•
•
•
•
Operations on Functions
Composition
Elementary Functions
Vertical and Horizontal Shifts
Reflections, Expansions, and Contractions
If two functions f and g are both defined at a real number x, and if f(x) and g(x) are
both real numbers, then it is possible to perform real number operations such as addition, subtraction, multiplication, or division with f(x) and g(x). Furthermore, if g(x) is
a number in the domain of f, then it is also possible to evaluate f at g(x). In this section we see how operations on the values of functions can be used to define operations on the functions themselves. We also investigate the graphic implications of
some of these operations.
• Operations
on Functions
The functions f and g given by
f(x) 2x 3
and
g(x) x2 4
are defined for all real numbers. Thus, for any real x we can perform the following
operations:
f(x) g(x) 2x 3 x2 4 x2 2x 1
f(x) g(x) 2x 3 (x2 4) x2 2x 7
f(x)g(x) (2x 3)(x2 4) 2x3 3x2 8x 12
166
2 Graphs and Functions
For x 2 we can also form the quotient
f(x)
2x 3
2
g(x)
x 4
x 2
Notice that the result of each operation is a new function. Thus, we have
( f g)(x) f(x) g(x) x2 2x 1
Sum
( f g)(x) f(x) g(x) x 2x 7
Difference
2
( fg)(x) f(x)g(x) 2x3 3x2 8x 12
Product
f(x)
2x 3
gf (x) g(x)
x 4
Quotient
2
x 2
Notice that the sum, difference, and product functions are defined for all values of x,
as were f and g, but the domain of the quotient function must be restricted to exclude
those values where g(x) 0.
DEFINITION 1
Operations on Functions
The sum, difference, product, and quotient of the functions f and g are the
functions defined by
( f g)(x) f(x) g(x)
Sum function
( f g)(x) f(x) g(x)
Difference function
( fg)(x) f(x)g(x)
f(x)
gf (x) g(x)
Product function
g(x) 0
Quotient function
Each function is defined on the intersection of the domains of f and g, with the
exception that the values of x where g(x) 0 must be excluded from the domain
of the quotient function.
EXAMPLE 1
Finding the Sum, Difference, Product, and Quotient Functions
Let f(x) 4 x and g(x) 3 x. Find the functions f g, f g, fg, and f /g,
and find their domains.
Solution
( f g)(x) f(x) g(x) 4 x 3 x
( f g)(x) f(x) g(x) 4 x 3 x
( fg)(x) f(x)g(x) 4 x3 x
(4 x)(3 x)
12 x x2
2-5 Combining Functions
167
f(x)
4x
4 x
gf (x) g(x)
3 x 3 x
The domains of f and g are
Domain of f
[
3
x
Domain of f: x 4 or (, 4]
4
0
Domain of g: x 3 or [3, )
Domain of g
[
3
x
4
0
The intersection of these domains is
(, 4] [3, ) [3, 4]
Domain of f g, f g, and fg
[
[
3
)
3
4
0
Domain of
f
g
[
0
x
This is the domain of the functions f g, f g, and fg. Since g(3) 0, x 3
must be excluded from the domain of the quotient function. Thus,
Domain of
x
4
Matched Problem 1
• Composition
f
: (3, 4]
g
Let f(x) x and g(x) 10 x. Find the functions f g, f g, fg, and f/g,
and find their domains.
Consider the function h given by the equation
h(x) 2x 1
Inside the radical is a first-degree polynomial that defines a linear function. So the
function h is really a combination of a square root function and a linear function. We
can see this more clearly as follows. Let
u 2x 1 g(x)
y u f(u)
Then
h(x) f [g(x)]
The function h is said to be the composite of the two functions f and g. (Loosely
speaking, we can think of h as a function of a function.) What can we say about the
domain of h given the domains of f and g? In forming the composite h(x) f [g(x)]:
x must be restricted so that x is in the domain of g and g(x) is in the domain of
f. Since the domain of f, where f(u) u, is the set of nonnegative real numbers,
we see that g(x) must be nonnegative; that is,
g(x) 0
2x 1 0
x 12
Thus, the domain of h is this restricted domain of g.
168
2 Graphs and Functions
A special function symbol is often used to represent the composite of two functions, which we define in general terms below.
DEFINITION 2
Composite Functions
Given functions f and g, then f g is called their composite and is defined by
the equation
( f g)(x) f [g(x)]
The domain of f g is the set of all real numbers x in the domain of g where
g(x) is in the domain of f.
As an immediate consequence of Definition 2, we have (see Fig. 1):
The domain of f g is always a subset of the domain of g, and the range
of f g is always a subset of the range of f.
FIGURE 1 Composite functions.
g
x
Domain g
EXAMPLE 2
(f g)(x) f [g (x)]
f
g
Domain f g
Range f g
f
g (x)
Range g
Domain f
Range f
Finding the Composition of Two Functions
Find ( f g)(x) and (g f )(x) and their domains for f(x) x10 and g(x) 3x 4 1.
Solution
( f g)(x) f [g(x)] f(3x4 1) (3x4 1)10
(g f )(x) g[ f (x)] g(x10) 3(x10)4 1 3x 40 1
The functions f and g are both defined for all real numbers. If x is any real number,
then x is in the domain of g, g(x) is in the domain of f, and, consequently, x is in the
domain of f g. Thus, the domain of f g is the set of all real numbers. Using similar reasoning, the domain of g f also is the set of all real numbers.
Matched Problem 2
Find ( f g)(x) and (g f )(x) and their domains for f (x) 2x 1 and g(x) (x 1)/2.
If two functions are both defined for all real numbers, then so is their
composition.
2-5 Combining Functions
EXPLORE-DISCUSS 1
169
Verify that if f(x) 1/(1 2x) and g(x) 1/x, then ( f g)(x) x/(x 2). Clearly,
f g is not defined at x 2. Are there any other values of x where f g is not
defined? Explain.
If either function in a composition is not defined for some real numbers, then, as
Example 3 illustrates, the domain of the composition may not be what you first think
it should be.
EXAMPLE 3
Finding the Composition of Two Functions
Find ( f g)(x) and its domain for f (x) 4 x2 and g(x) 3 x.
Solution
We begin by stating the domains of f and g, a good practice in any composition
problem:
Domain f : 2 x 2
or
Domain g: x 3
(, 3]
or
[2, 2]
Next we find the composition:
( f g)(x) f [g(x)] f(3 x)
4 (3 x)2
4 (3 x)
(t)2 t, t 0
1 x
Even though 1 x is defined for all x 1, we must restrict the domain of f g
to those values that also are in the domain of g. Thus,
Domain f g: x 1 and x 3
Matched Problem 3
CAUTION
or
[1, 3]
Find ( f g)(x) and its domain for f (x) 9 x2 and g(x) x 1.
The domain of f g cannot always be determined simply by examining the
final form of ( f g)(x). Any numbers that are excluded from the domain of g
must also be excluded from the domain of f g.
In calculus, it is not only important to be able to find the composition of two
functions, but also to recognize when a given function is the composition of two
simpler functions.
170
2 Graphs and Functions
EXAMPLE 4
Recognizing Composition Forms
Express h as a composition of two simpler functions for
h(x) (3x 5)5
Solution
If we let f(x) x5 and g(x) 3x 5, then
h(x) (3x 5)5 f (3x 5) f [g(x)] ( f g)(x)
and we have expressed h as the composition of f and g.
Matched Problem 4
• Elementary
Express h as a composition of the square root function and a linear function for
h(x) 4x 7.
The functions
Functions
g(x) x2 4
h(x) (x 4)2
k(x) 4x2
can all be obtained from the function f(x) x2 by performing simple operations on f:
g(x) f (x) 4
h(x) f (x 4)
k(x) 4f (x)
It follows that the graphs of functions g, h, and k are closely related to the graph of
function f. Before exploring relationships of this type, we want to identify some elementary functions, summarize their basic properties, and include them in our library
of elementary functions. Figure 2 shows six basic functions that you will encounter
frequently. You should know the definition, domain, and range of each and be able to
sketch their graphs.
f (x)
FIGURE 2 Some basic functions
and their graphs. [ Note: Letters used
to designate these functions may
vary from context to context; R is the
set of all real numbers.]
g(x)
h(x)
5
5
5
5
5
x
5
5
x
5
5
5
Identity function
f (x) x
Domain: R
Range: R
(a)
Absolute value function
g(x) x
Domain: R
Range: [0, )
(b)
Square function
h(x) x 2
Domain: R
Range: [0, )
(c)
x
2-5 Combining Functions
m(x)
n (x)
5
5
5
Horizontal Shifts
EXPLORE-DISCUSS 2
x
5
5
5
Cube function
m(x) x 3
Domain: R
Range: R
(d)
• Vertical and
p (x)
5
5
Square root function
n(x) x
Domain: [0, )
Range: [0, )
(e)
x
5
5
x
5
Cube root function
3
p(x) x
Domain: R
Range: R
(f)
How are the graphs of y ( f g)(x), y ( fg)(x), and y ( f g)(x) related to the
graphs of y f(x) and y g(x)? In general, this is a difficult question to answer.
However, if g is chosen to be a very simple function, such as g(x) k or g(x) x h, then we can establish some very useful relationships between the graph of
y f(x) and the graphs of y f (x) k, y kf(x), and y f(x h). We refer to
the graph obtained by performing one of these operations on a function f as a transformation of the graph of y f (x).
Let f(x) x .
(A) Graph y f (x) k for k 2, 0, and 1 simultaneously in the
dinate system. Describe the relationship between the graph of y
the graph of y f(x) k for k, any real number.
(B) Graph y f (x h) for h 2, 0, and 1 simultaneously in the
dinate system. Describe the relationship between the graph of y
the graph of y f(x h) for h, any real number.
EXAMPLE 5
171
same coor f (x) and
same coor f(x) and
Vertical and Horizontal Shifts
(A) How are the graphs of y x2 2 and y x2 3 related to the graph of
y x2? Confirm your answer by graphing all three functions simultaneously in
the same coordinate system.
(B) How are the graphs of y (x 2)2 and y (x 3)2 related to the graph of
y x2? Confirm your answer by graphing all three functions simultaneously in
the same coordinate system.
Solutions
(A) The graph of y x2 2 is the same as the graph of y x2 shifted upward 2
units, and the graph of y x2 3 is the same as the graph of y x2 shifted
downward 3 units. Figure 3 on the next page confirms these conclusions. [It
appears that the graph of y f(x) k is the graph of y f(x) shifted up if
k is positive and down if k is negative.]
172
2 Graphs and Functions
y
FIGURE 3 Vertical shifts.
5
y x2 2
y x2
x
5
5
y x2 3
5
(B) The graph of y (x 2)2 is the same as the graph of y x2 shifted to the left
2 units, and the graph of y (x 3)2 is the same as the graph of y x2 shifted
to the right 3 units. Figure 4 confirms these conclusions. [It appears that the
graph of y f (x h) is the graph of y f (x) shifted right if h is negative and
left if h is positive—the opposite of what you might expect.]
y
y x2
FIGURE 4 Horizontal shifts.
y (x 3)2
5
y (x 2)2
5
Matched Problem 5
5
x
(A) How are the graphs of y x 3 and y x 1 related to the graph of
y x? Confirm your answer by graphing all three functions simultaneously in
the same coordinate system.
(B) How are the graphs of y x 3 and y x 1 related to the graph of
y x? Confirm your answer by graphing all three functions simultaneously in
the same coordinate system.
Comparing the graph of y f(x) k with the graph of y f(x), we see that the
graph of y f(x) k can be obtained from the graph of y f (x) by vertically translating (shifting) the graph of the latter upward k units if k is positive and downward
k units if k is negative. Comparing the graph of y f (x h) with the graph of
y f(x), we see that the graph of y f(x h) can be obtained from the graph of
y f(x) by horizontally translating (shifting) the graph of the latter h units to the
left if h is positive and h units to the right if h is negative.
EXAMPLE 6
Vertical and Horizontal Translations (Shifts)
The graphs in Figure 5 are either horizontal or vertical shifts of the graph of f(x) x .
Write appropriate equations for functions H, G, M, and N in terms of f.
2-5 Combining Functions
y
FIGURE 5 Vertical and horizontal
shifts.
173
y
G
M
f
5
f
H
5
N
5
5
x
5
x
5
5
Solution
Functions H and G are vertical shifts given by
G(x) x 1
H(x) x 3
Functions M and N are horizontal shifts given by
N(x) x 3
M(x) x 2
Matched Problem 6
The graphs in Figure 6 are either horizontal or vertical shifts of the graph of
f(x) x3. Write appropriate equations for functions H, G, M, and N in terms of f.
G y
FIGURE 6 Vertical and horizontal
shifts.
y MfN
f H
5
5
• Reflections,
Expansions,
and Contractions
EXPLORE-DISCUSS 3
5
5
x
5
5
x
We now investigate how the graph of y Af(x) is related to the graph of y f(x)
for different real numbers A.
(A) Graph y Ax for A 1, 2, and 12 simultaneously in the same coordinate
system.
(B) Graph y Ax for A 1, 2, and 12 simultaneously in the same coordinate system.
(C) Describe the relationship between the graph of h(x) x and the graph of
G(x) Ax for A any real number.
174
2 Graphs and Functions
Comparing the graph of y Af (x) with the graph of y f (x), we see that the
graph of y Af(x) can be obtained from the graph of y f (x) by multiplying each
ordinate value of the latter by A. The result is a vertical expansion of the graph of
y f(x) if A 1, a vertical contraction of the graph of y f (x) if 0 A 1, and
a reflection in the x axis if A 1.
EXAMPLE 7
Reflections, Expansions, and Contractions
3
3
3
(A) How are the graphs of y 2
x and y 0.5
x related to the graph of y x?
Confirm your answer by graphing all three functions simultaneously in the same
coordinate system.
3
3
(B) How is the graph of y 2
x related to the graph of y x? Confirm your
answer by graphing both functions simultaneously in the same coordinate system.
Solution
3
3
(A) The graph of y 2
x is a vertical expansion of the graph of y x by a fac3
tor of 2, and the graph of y 0.5x is a vertical contraction of the graph of
3
y
x by a factor of 0.5. Figure 7 confirms this conclusion.
y
FIGURE 7 Vertical expansion and
contraction.
5
3
y 2x
3
y x
3
y 0.5x
5
5
x
5
3
(B) The graph of y 2
x is a reflection in the x axis and a vertical expansion of
3
the graph of y x. Figure 8 confirms this conclusion.
y
FIGURE 8 Reflection and vertical
expansion.
5
3
y x
5
5
x
3
y 2x
5
Matched Problem 7
(A) How are the graphs of y 2x and y 0.5x related to the graph of y x? Confirm your answer by graphing all three functions simultaneously in the same
coordinate system.
(B) How is the graph of y 0.5x related to the graph of y x? Confirm your
answer by graphing both functions in the same coordinate system.
2-5 Combining Functions
175
The various transformations considered above are summarized in the following
box for easy reference:
Graph Transformations (Summary)
Vertical Translation [see Fig. 9(a)]:
k0
k 0
y f(x) k
Shift graph of y f(x) up k units
Shift graph of y f(x) down k units
Horizontal Translation [see Fig. 9(b)]:
h0
h0
y f(x h)
Shift graph of y f(x) left h units
Shift graph of y f(x) right h units
Reflection [see Fig. 9(c)]:
y f(x)
Reflect graph of y f(x) in the x axis
Vertical Expansion and Contraction [see Fig. 9(d)]:
y Af(x)
A1
Vertically expand graph of y f(x)
by multiplying each ordinate value by A
0A1
Vertically contract graph of y f(x)
by multiplying each ordinate value by A
FIGURE 9 Graph transformations.
y
g y
f
y
h
y
g
f
5
5
5
g
f
5
f
5
5
x
h
5
5
x
5
5
h
x
5
5
g(x) f (x) 2
h(x) f (x) 3
(a) Vertical translation
EXPLORE-DISCUSS 4
5
g(x) f (x 3)
h(x) f (x 2)
(b) Horizontal translation
5
5
g
g(x) f (x)
(c) Reflection
g(x) 2f (x)
h(x) 0.5f (x)
(d) Expansion and contraction
Use a graphing utility to explore the graph of y A(x h)2 k for various values of the constants A, h, and k. Discuss how the graph of y A(x h)2 k is
related to the graph of y x2.
x
176
2 Graphs and Functions
EXAMPLE 8
Combining Graph Transformations
The graph of y g(x) in Figure 10 is a transformation of the graph of y x2. Find
an equation for the function g.
y
FIGURE 10
5
y g(x)
5
Solution
5
x
To transform the graph of y x2 [Fig. 11(a)] into the graph of y g(x), we first
reflect the graph of y x2 in the x axis [Fig. 11(b)], then shift it to the right two units
[Fig. 11(c)]. Thus, an equation for the function g is
g(x) (x 2)2
y
FIGURE 11
y
5
5
5
y x2
5
5
x
5
y x 2
5
5
x
5
y (x 2)2
x
5
5
y x2
(a)
Matched Problem 8
y
y x 2
(b)
y (x 2)2
(c)
The graph of y h(x) in Figure 12 is a transformation of the graph of y x3. Find
an equation for the function h.
y
FIGURE 12
5
y h(x)
5
5
5
x
2-5 Combining Functions
177
Answers to Matched Problems
1. ( f g)(x) x 10 x, ( f g)(x) x 10 x, ( fg)(x) 10x x2,
( f/g)(x) x/(10 x); the functions f g, f g, and fg have domain [0, 10], the domain of f/g is
[0, 10)
2. ( f g)(x) x, domain (, )
(g f )(x) x, domain (, )
3. ( f g)(x) 10 x; domain: x 1 and x 10 or [1, 10]
4. h(x) ( f g)(x), where f (x) x and g(x) 4x 7
5. (A) The graph of y x 3 is the same as the graph of y x shifted upward 3 units, and the
graph of y x 1 is the same as the graph of y x shifted downward 1 unit. The figure
confirms these conclusions.
(B) The graph of y x 3 is the same as the graph of y x shifted to the left 3 units, and
the graph of y x 1 is the same as the graph of y x shifted to the right 1 unit. The
figure confirms these conclusions.
y
y
y x 3
5
5
5
y x
y x 1
x
5
y x 3
y x
y x 1
5
5
x
5
5
6. G(x) (x 3)3, H(x) (x 1)3, M(x) x3 3, N(x) x3 4
7. (A) The graph of y 2x is a vertical expansion of the graph of y x, and the graph of y 0.5x is
a vertical contraction of the graph of y x. The figure confirms these conclusions.
(B) The graph of y 0.5x is a vertical contraction and a reflection in the x axis of the graph of
y x. The figure confirms this conclusion.
y
y 2x
y
yx
5
yx
5
y 0.5x
5
5
x
5
5
x
y 0.5x
5
5
8. The graph of function h is a reflection in the x axis and a horizontal translation of 3 units to the left
of the graph of y x3. An equation for h is h(x) (x 3)3.
EXERCISE
2-5
A
Without looking back in the text, indicate the domain and
range of each of the following functions. (Making rough
sketches on scratch paper may help.)
1. h(x) x
4. f(x) 0.5 x
3
2. m(x) x
5. F(x) 0.5x
3
In Problems 7–10, for the indicated functions f and g, find
the functions f g, f g, fg, and f/g, and find their
domains.
7. f(x) 4x; g(x) x 1
3. g(x) 2x2
8. f(x) 3x; g(x) x 2
6. G(x) 4x
9. f(x) 2x2; g(x) x2 1
3
178
2 Graphs and Functions
10. f(x) 3x; g(x) x2 4
In Problems 35–40, for the indicated functions f and g, find
the functions f g and g f, and find their domains.
In Problems 11–14, for the indicated functions f and g, find
the functions f g and g f, and find their domains.
11. f(x) x2 3;
35. f (x) x 2; g(x) 4 x
36. f (x) x 1; g(x) x 2
g(x) x2 4x
12. f(x) x2 5x; g(x) x2 1
37. f (x) x 3; g(x) 13. f(x) 2x2/3; g(x) x3 1
1
x2
x
; g(x) 2 x
x4
14. f(x) 4 x3; g(x) 3x1/3
38. f (x) Problems 15–22 refer to the functions f and g given by the
graphs below (the domain of each function is [2, 2]).
39. f (x) x 2 ; g(x) Use the graph of f or g, as required, to graph each given
function.
40. f (x) f(x)
g (x)
5
5
5
5
x
5
5
5
x
x
x3
x
; g(x) x 3
x4
Each graph in Problems 41–46 is the result of applying a
sequence of transformations to the graph of one of the six
basic functions in Figure 2. Identify the basic function and
describe the transformation verbally. Write an equation for
the given graph.
Check your equations in Problems 41–46 by graphing each
on a graphing utility.
41.
5
y
5
15. f(x) 2
16. g(x) 1
17. g(x 2)
18. f(x 1)
19. f (x)
20. g(x)
21. 2g(x)
22.
1
2 f (x)
5
B
5
x
5
In Problems 23–28, indicate how the graph of each function
is related to the graph of one of the six basic functions in
Figure 2. Sketch a graph of each function.
42.
y
5
Check your descriptions and graphs in Problems 23–28 by
graphing each function on a graphing utility.
23. g(x) x 2
24. h(x) x 4
25. f(x) (x 2) 4
26. m(x) (x 1) 3
27. f (x) 4 2x
3
28. g(x) 2 3
x
2
5
2
In Problems 29–34, for the indicated functions f and g, find
the functions f g, f g, fg, and f/g, and find their
domains.
5
43.
x
y
29. f (x) x 2; g(x) 4 x
5
30. f (x) 5 x; g(x) x 1
31. f (x) 5 2x; g(x) 3 x
32. f (x) 3x 6; g(x) x 1
5
5
33. f (x) x2 x 2; g(x) 24 2x x2
34. f (x) x2 3x 10; g(x) x2 x 12
5
x
2-5 Combining Functions
44.
y
5
5
5
x
179
Changing the order in a sequence of transformations may
change the final result. Investigate each pair of transformations in Problems 53–58 to determine whether reversing
their order can produce a different result. Support your
conclusions with specific examples and/or mathematical
arguments.
53. Vertical shift, horizontal shift.
54. Vertical shift, reflection in y axis.
5
45.
55. Vertical shift, reflection in x axis.
56. Vertical shift, vertical expansion.
y
57. Horizontal shift, reflection in x axis.
5
58. Horizontal shift, vertical contraction.
5
5
x
5
46.
y
5
In Problems 59–66, express h as a composition of two
simpler functions f and g of the form f(x) xn and
g(x) ax b, where n is a rational number and a and b
are integers.
59. h(x) (2x 7)4
60. h(x) (3 5x)7
61. h(x) 4 2x
62. h(x) 3x 11
63. h(x) 3x7 5
64. h(x) 5x6 3
65. h(x) 5
5
x
5
In Problems 47–52, the graph of the function g is formed by
applying the indicated sequence of transformations to the
given function f. Find an equation for the function g and
graph g using 5 x 5 and 5 y 5.
47. The graph of f (x) x is shifted 2 units to the left and 3
units up.
4
3
x
66. h(x) C
Each of the following graphs involves a reflection in the x
axis and/or a vertical expansion or contraction of one of the
basic functions in Figure 2. Identify the basic function and
describe the transformation verbally. Write an equation for
the given graph.
Check your equations in Problems 67–70 by graphing each
on a graphing utility.
67.
y
5
48. The graph of f (x) x is shifted 3 units to the right and 2
units down.
3
49. The graph of f (x) x is reflected in the x axis and shifted
to the right 3 units.
50. The graph of f (x) x is reflected in the x axis and shifted
to the left 1 unit.
51. The graph of f(x) x3 is reflected in the x axis and shifted
to the left 2 units and up 1 unit.
52. The graph of f(x) x2 is reflected in the x axis and shifted
to the right 2 units and down 4 units.
2
1
x
5
5
5
x
180
2 Graphs and Functions
68.
In Problems 79–84, for the indicated functions f and g, find
the functions f g and g f, and find their domains.
y
5
79. f (x) x2; g(x) 9 x
80. f (x) x 16; g(x) x2
5
5
x
5
69.
81. f (x) x2
2x 1
; g(x) x2
x3
82. f (x) x2
x5
; g(x) x4
x1
83. f (x) x2 5; g(x) x2 4
y
84. f (x) x2 8; g(x) x2 9
5
5
5
x
APPLICATIONS
85. Market Research. The demand x and the price p (in dollars) for a certain product are related by
5
70.
x f( p) 4,000 200p
y
The revenue (in dollars) from the sale of x units is given by
5
R(x) 20x 5
5
x
1 2
x
200
and the cost (in dollars) of producing x units is given by
C(x) 10x 30,000
5
Express the profit as a function of the price p.
71. Are the functions fg and gf identical? Justify your answer.
72. Are the functions f g and g f identical? Justify your answer.
73. Is there a function g that satisfies f g g f f for all
functions f? If so, what is it?
74. Is there a function g that satisfies fg gf f for all functions f? If so, what is it?
In Problems 75–78, for the indicated functions f and g, find
the functions f g, f g, fg, and f/g, and find their
domains.
1
75. f (x) x ;
x
g(x) x 76. f (x) x 1; g(x) x 77. f (x) 1 1
x
6
x1
x
x
; g(x) 1 x
x
78. f (x) x x ; g(x) x x
86. Market Research. The demand x and the price p (in dollars) for a certain product are related by
x f( p) 5,000 100p
The revenue (in dollars) from the sale of x units and the cost
(in dollars) of producing x units are given, respectively, by
R(x) 50x 1 2
x
100
and
C(x) 20x 40,000
Express the profit as a function of the price p.
87. Family of Curves. In calculus, solutions to certain types of
problems often involve an unspecified constant. For example, consider the equation
y
1 2
x C
C
where C is a positive constant. The collection of graphs of
this equation for all permissible values of C is called a family of curves. On the same axes, graph the members of this
family corresponding to C 1, 2, 3, and 4.
2-5 Combining Functions
88. Family of Curves. A family of curves is defined by the
equation
y 2C ★
2 2
x
C
where C is a positive constant. On the same axes, graph the
members of this family corresponding to C 1, 2, 3, and 4.
91. Fluid Flow. A conical paper cup with diameter 4 inches and
height 4 inches is initially full of water. A small hole is
made in the bottom of the cup and the water begins to flow
out of the cup. Let h and r be the height and radius, respectively, of the water in the cup t minutes after the water begins to flow.
4 inches
89. Fluid Flow. A cubic tank is 4 feet on a side and is initially
full of water. Water flows out an opening in the bottom of
the tank at a rate proportional to the square root of the depth
(see the figure). Using advanced concepts from mathematics and physics, it can be shown that the volume of the water in the tank t minutes after the water begins to flow is
given by
V(t) 64
(C t)2
C2
r
4 inches
h
0tC
where C is a constant that depends on the size of the opening. Graph V(t) for C 1, C 2, C 4, and C 8.
4 feet
181
1
V 3 r 2h
(A) Express r as a function of h.
(B) Express the volume V as a function of h.
(C) If the height of the water after t minutes is given by
4 feet
h(t) 0.5t
4 feet
express V as a function of t.
★
90. Evaporation. A water trough with triangular ends is 9 feet
long, 4 feet wide, and 2 feet deep (see the figure). Initially,
the trough is full of water, but due to evaporation, the volume of the water in the trough decreases at a rate proportional to the square root of the volume. Using advanced
concepts from mathematics and physics, it can be shown
that the volume after t hours is given by
92. Evaporation. A water trough with triangular ends is 6 feet
long, 4 feet wide, and 2 feet deep. Initially, the trough is full
of water, but due to evaporation, the volume of the water is
decreasing. Let h and w be the height and width, respectively, of the water in the tank t hours after it began to evaporate.
4 feet
6 feet
V(t) 1
(t 6C)2
C2
0t6C
2 feet
where C is a constant. Graph V(t) for C 4, C 5, and
C 6.
4 feet
9 feet
2 feet
w
h
V 3wh
(A) Express w as a function of h.
(B) Express V as a function of h.
(C) If the height of the water after t hours is given by
h(t) 2 0.2t
express V as a function of t.