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84 Chapter 1 Functions and Their Graphs
Arithmetic Combinations of Functions
Just as two real numbers can be combined by the operations of addition, subtrac-
tion, multiplication, and division to form other real numbers, two functions can be
combined to create new functions. For example, the functions given by
and can be combined to form the sum, difference,
product, and quotient of and
Sum
Difference
Product
Quotient
The domain of an arithmetic combination of functions and consists of all
real numbers that are common to the domains of and In the case of the quo-
tient there is the further restriction that
Finding the Sum of Two Functions
Given and find
Solution
Now try Exercise 5(a).
f g x f x g x 2 x 1 x^2 2 x 1 x^2 4 x
f x 2 x 1 g x x^2 2 x 1, f g x .
f x g x , g x 0.
f g.
f g
f x
g x
2 x 3
x^2 1
, x ± 1
2 x^3 3 x^2 2 x 3
f x g x 2 x 3 x^2 1
x^2 2 x 2
f x g x 2 x 3 x^2 1
x^2 2 x 4
f x g x 2 x 3 x^2 1
f g.
f x 2 x 3 g x x^2 1
What you should learn
- Add, subtract, multiply, and divide functions.
- Find the composition of one function with another function.
- Use combinations and compo- sitions of functions to model and solve real-life problems.
Why you should learn it Compositions of functions can be used to model and solve real-life problems. For instance, in Exercise 68 on page 92, compositions of functions are used to determine the price of a new hybrid car.
Combinations of Functions: Composite Functions
© Jim West/The Image Works
1.
Sum, Difference, Product, and Quotient of Functions
Let and be two functions with overlapping domains. Then, for all
common to both domains, the sum , difference , product , and quotient of
and are defined as follows.
1. Sum:
2. Difference:
3. Product:
4. Quotient: (^) g x 0
f
g ^
x
f x
g x
fg x f x (^) g x
f g x f x g x
f g x f x g x
g
f
f g x
Example 1
Section 1.8 Combinations of Functions: Composite Functions 85
Finding the Difference of Two Functions
Given and find Then evaluate
the difference when
Solution
The difference of and is
When the value of this difference is
Now try Exercise 5(b).
In Examples 1 and 2, both and have domains that consist of all real
numbers. So, the domains of and are also the set of all real
numbers. Remember that any restrictions on the domains of and must be
considered when forming the sum, difference, product, or quotient of and
Finding the Domains of Quotients of Functions
Find and for the functions given by
and
Then find the domains of
Solution
The quotient of and is
and the quotient of and is
The domain of is and the domain of is The intersection of
these domains is So, the domains of and are as follows.
Note that the domain of includes but not because
yields a zero in the denominator, whereas the domain of includes
but not because yields a zero in the denominator.
Now try Exercise 5(d).
x 0, x 0
g f x 2,
f g x 0, x 2, x 2
Domain of (^)
g
f
Domain of (^) : 0, 2
f
g
g
f
f
g
f 0, (^) g 2, 2.
g
f ^
x
g x
f x
4 x^2
x
g f
f
g ^
x
f x
g x
x
4 x^2
f g
f g and g f.
f x x g x 4 x^2.
g
f ^ x
f
g ^
x
f g.
f g
f g f g
f g
f g 2 2 ^2 2
x 2,
x^2 2.
2 x 1 x^2 2 x 1
f g x f x g x
f g
x 2.
Additional Examples f x 2 x 1 g x x^2 2 x 1, f g x .
a. Given and find Solution
b. Given and
find
Solution
x 1 ,^ x^ ^0
(^)
x x 1
x
gf x gx f x
gf x.
gx x x 1
f x 1 , x
3 x^2 15 x
x 5 3 x
fgx f x gx
fgx.
f x x 5 gx 3 x,
Example 2
Example 3
Section 1.8 Combinations of Functions: Composite Functions 87
Finding the Domain of a Composite Function
Given and find the composition Then
find the domain of
Solution
From this, it might appear that the domain of the composition is the set of all real
numbers. This, however is not true. Because the domain of is the set of all real
numbers and the domain of is the domain of is
Now try Exercise 35.
In Examples 4 and 5, you formed the composition of two given functions. In
calculus, it is also important to be able to identify two functions that make up a
given composite function. For instance, the function given by
is the composition of with where and That is,
Basically, to “decompose” a composite function, look for an “inner” function and
an “outer” function. In the function above, is the inner function
and is the outer function.
Decomposing a Composite Function
Write the function given by as a composition of two functions.
Solution
One way to write as a composition of two functions is to take the inner func-
tion to be and the outer function to be
Then you can write
Now try Exercise 47.
h x
x 2 ^2
x 2 ^2 f x 2 f g x .
f x
x^2
x ^2.
g x x 2
h
h x
x 2 ^2
f x x^3
h g x 3 x 5
h x 3 x 5 ^3 g x ^3 f g x .
f g , f x x^3 g x 3 x 5.
h x 3 x 5 ^3
h
3 ≤ x ≤ 3.
g 3 ≤ x ≤ 3, f g
f
x^2
9 x^2 9
9 x^2 ^2 9
f 9 x^2
f g x f g x
f g .
f x x^2 9 g x 9 x^2 , f g x .
Activities
- Given and find Answer:
- Given the functions
and
find the composition of with Then find the domain of the composition. Answer: The
domain of is the set of all non- negative real numbers except
- Find two functions and such that (There are many correct answers.)
a.
Answer: and
b. Answer: and gx 2 x 3
f x x 4
hx 2 x 3 ^4
gx 3 x 1
f x 1 x
hx
3 x 1
f (^) gx hx.
f g
x 4.
f (^) g
f (^) gx
x 2.
f g.
f x 1 gx x , x 2
f (^) gx 12 x^2 2
f (^) g.
f x 3 x^2 2 gx 2 x,
You can use a graphing utility to determine the domain of a composition of functions. For the composition in Example 5, enter the function composition as
You should obtain the graph shown below. Use the trace feature to determine that the x-coordinates of points on the graph extend from to 3. So, the domain of is
− 5
− 10
5
1
f (^) gx 3 ≤ x ≤ 3.
y 9 x^2
2 9.
Te c h n o l o g y Example 5
Example 6
88 Chapter 1 Functions and Their Graphs
Application
Bacteria Count
The number of bacteria in a refrigerated food is given by
where is the temperature of the food in degrees Celsius. When the food is
removed from refrigeration, the temperature of the food is given by
where is the time in hours. (a) Find the composition and interpret its
meaning in context. (b) Find the time when the bacterial count reaches 2000.
Solution
a.
The composite function represents the number of bacteria in the food
as a function of the amount of time the food has been out of refrigeration.
b. The bacterial count will reach 2000 when Solve this
equation to find that the count will reach 2000 when hours. When you
solve this equation, note that the negative value is rejected because it is not in
the domain of the composite function.
Now try Exercise 65.
t 2.
320 t^2 420 2000.
N T t
320 t^2 420
320 t^2 320 t 80 320 t 160 500
20 16 t^2 16 t 4 320 t 160 500
N T t 20 4 t 2 ^2 80 4 t 2 500
t N T t
T t 4 t 2, 0 ≤ t ≤ 3
T
N T 20 T^2 80 T 500, 2 ≤ T ≤ 14
N
Writing About Mathematics To expand on this activity, you might consider asking your students to use the tables they created in parts (a) and (b), along with a table of values for and to demonstrate and explain how the tables can be manipulated to yield tables of values for hxand gx.
f x,
x
WRITING ABOUT^ MATHEMATICS
Analyzing Arithmetic Combinations of Functions
a. Use the graphs of and in Figure 1.91 to make a table showing the values of when , and 6. Explain your reasoning. b. Use the graphs of and in Figure 1.91 to make a table showing the values of hxwhen x 1, 2, 3, 4, 5, and 6. Explain your reasoning.
f f h
gx x 1, 2, 3, 4, 5
f f g
x 1 2 3 4 5 6
1
2
3
4
5
6 f
y
FIGURE 1.
x 1 2 3 4 5 6
1
2
3
4
5
6 f + g
y
x 1 2 3 4 5 6
1
2
3
4
5
6 f − h
y
Example 7
90 Chapter 1 Functions and Their Graphs
In Exercises 35–42, find (a) and (b) Find the domain of each function and each composite function.
**35.
40.**
41.
In Exercises 43–46, use the graphs of and to evaluate the functions.
43. (a) (b) 44. (a) (b) 45. (a) (b) 46. (a) (b)
In Exercises 47–54, find two functions and such that (There are many correct answers.)
**47. 48.
- 52.**
55. Stopping Distance The research and development department of an automobile manufacturer has determined that when a driver is required to stop quickly to avoid an accident, the distance (in feet) the car travels during the driver’s reaction time is given by where is the speed of the car in miles per hour. The distance (in feet) traveled while the driver is braking is given by Find the function that represents the total stopping distance Graph the functions and on the same set of coor- dinate axes for 56. Sales From 2000 to 2005, the sales (in thousands of dollars) for one of two restaurants owned by the same parent company can be modeled by
where represents 2000. During the same six-year period, the sales (in thousands of dollars) for the second restaurant can be modeled by
(a) Write a function that represents the total sales of the two restaurants owned by the same parent company. (b) Use a graphing utility to graph and in the same viewing window.
57. Vital Statistics Let be the number of births in the United States in year and let represent the number of deaths in the United States in year where corresponds to 2000. (a) If is the population of the United States in year find the function that represents the percent change in the population of the United States. (b) Interpret the value of 58. Pets Let be the number of dogs in the United States in year and let be the number of cats in the United States in year where corresponds to 2000. (a) Find the function that represents the total number of dogs and cats in the United States. (b) Interpret the value of (c) Let represent the population of the United States in year where corresponds to 2000. Find and interpret 59. Military Personnel The total numbers of Army personnel (in thousands) and Navy personnel (in thousands) from 1990 to 2002 can be approximated by the models
and
where represents the year, with corresponding to
- (Source: Department of Defense) (a) Find and interpret Evaluate this function for 8, and 12. (b) Find and interpret Evaluate this function for t 4,8, and 12.
A N t .
t 4,
A N t .
t t 0
N t 1.95 t^2 42.2 t 603
A t 3.36 t^2 59.8 t 735
A N
h t
p t n t
t , t 0
n t
p 5 .
p t
t , t 0
t , c t
d t
c 5 .
c t
p t t ,
t , t 0
t , d t
b t
R 1 , R 2 , R 3
R 3
R 2 254 0.78 t , t 0, 1, 2, 3, 4, 5.
R 2
t 0
R 1 480 8 t 0.8 t^2 , t 0, 1, 2, 3, 4, 5
R 1
0 ≤ x ≤ 60.
T. R , B , T
B x 151 x^2.
R x 34 x , x
h x
27 x^3 6 x 10 27 x^3
h x
x^2 3 4 x^2
h x
5 x 2 ^2
h x
x 2
h x ^3 x^2 4 h x 9 x
h x 2 x 1 ^2 h x 1 x ^3
f (^) g x h x.
f g
f (^) g 1 g (^) f 3
f (^) g 2 g (^) f 2
f g 1 fg 4
f g 3 f g 2
x 1 2 3 4
3
1
2
4 y^ =^ g ( x )
y
x 1 2 3 4
2
3
1
4
y y = f ( x )
f g
f x g x x 1
x^2 1
f x g x x 3
x
f x x 4 , g x 3 x
f x x , g x x 6
f x x^2 ^3 , g x x^6
f x x^2 1, g x x
f x ^3 x 5, g x x^3 1
f x x 4, g x x^2
f (^) g g (^) f.
Section 1.8 Combinations of Functions: Composite Functions 91
60. Sales The sales of exercise equipment (in millions of dollars) in the United States from 1997 to 2003 can be approximated by the function
and the U.S. population (in millions) from 1997 to 2003 can be approximated by the function
where represents the year, with corresponding to
- (Source: National Sporting Goods Association, U.S. Census Bureau)
(a) Find and interpret
(b) Evaluate the function in part (a) for 10, and 12.
62. Graphical Reasoning An electronically controlled ther- mostat in a home is programmed to lower the temperature automatically during the night. The temperature in the house (in degrees Fahrenheit) is given in terms of the time in hours on a 24-hour clock (see figure).
(a) Explain why is a function of (b) Approximate and (c) The thermostat is reprogrammed to produce a temper- ature for which How does this change the temperature? (d) The thermostat is reprogrammed to produce a temper- ature for which How does this change the temperature? (e) Write a piecewise-defined function that represents the graph.
63. Geometry A square concrete foundation is prepared as a base for a cylindrical tank (see figure).
(a) Write the radius of the tank as a function of the length of the sides of the square. (b) Write the area of the circular base of the tank as a function of the radius (c) Find and interpret A (^) r x .
r.
A
x
r
x
r
H H t T t 1.
H H t T t 1 .
T 4 T 15 .
T t.
t 3 6 912 15 18 2124
50
60
70
80
Time (in hours)
T
Temperature (in ˚F)
T t ,
t 7,
h t
E t P t
t t 7
P t 3.02 t 252.
P
E t 25.95 t^2 231.2 t 3356
E
61. Health Care Costs The table shows the total amounts (in billions of dollars) spent on health services and supplies in the United States (including Puerto Rico) for the years 1995 through 2001. The variables and represent out-of-pocket payments, insurance premiums, and other types of payments, respectively. (Source: Centers for Medicare and Medicaid Services)
(a) Use the regression feature of a graphing utility to find a linear model for and quadratic models for and Let represent 1995. (b) Find What does this sum represent? (c) Use a graphing utility to graph and in the same viewing window. (d) Use the model from part (b) to estimate the total amounts spent on health services and supplies in the years 2008 and 2010.
y 1 y 2 y 3
y 1 , y 2 , y 3 ,
y 1 y 2 y 3.
y 2 y 3. t 5
y 1
y 3
y 1 , y 2 ,
Model It
Year
y 1 y 2 y 3
