In some cases, it is necessary to decompose a complicated function. In other words, we can write it as a composition of two simpler functions. There may be more than one way to decompose a composite function, so we may choose the decomposition that appears to be most expedient.
Algebraic
5.
Given and find and Determine the domain for each function in interval notation.
6.
Given and find and Determine the domain for each function in interval notation.
7.
Given and find and Determine the domain for each function in interval notation.
8.
Given and find and Determine the domain for each function in interval notation.
9.
Given and find and Determine the domain for each function in interval notation.
10.
Given and find Determine the domain of the function in interval notation.
11.
Given and find the following:
- ⓐ
- ⓑ
- ⓒ
- ⓓ
- ⓔ
For the following exercises, use each pair of functions to find and Simplify your answers.
12.
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17.
For the following exercises, use each set of functions to find Simplify your answers.
18.
and
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and
20.
Given and find the following:
- ⓐ
- ⓑ the domain of in interval notation
- ⓒ
- ⓓ the domain of
- ⓔ
21.
Given and find the following:
- ⓐ
- ⓑ the domain of in interval notation
22.
Given the functions find the following:
- ⓐ
- ⓑ
23.
Given functions and state the domain of each of the following functions using interval notation:
- ⓐ
- ⓑ
- ⓒ
24.
Given functions and state the domain of each of the following functions using interval notation.
- ⓐ
- ⓑ
- ⓒ
25.
For and write the domain of in interval notation.
For the following exercises, find functions and so the given function can be expressed as
26.
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41.
Graphical
For the following exercises, use the graphs of shown in Figure 4, and shown in Figure 5, to evaluate the expressions.
For the following exercises, use graphs of shown in Figure 6, shown in Figure 7, and shown in Figure 8, to evaluate the expressions.
50.
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57.
Numeric
For the following exercises, use the function values for shown in Table 3 to evaluate each expression.
|
|
|
0 |
7 |
9 |
1 |
6 |
5 |
2 |
5 |
6 |
3 |
8 |
2 |
4 |
4 |
1 |
5 |
0 |
8 |
6 |
2 |
7 |
7 |
1 |
3 |
8 |
9 |
4 |
9 |
3 |
0 |
Table 3
58.
59.
60.
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62.
63.
64.
65.
For the following exercises, use the function values for shown in Table 4 to evaluate the expressions.
|
|
|
-3 |
11 |
-8 |
-2 |
9 |
-3 |
-1 |
7 |
0 |
0 |
5 |
1 |
1 |
3 |
0 |
2 |
1 |
-3 |
3 |
-1 |
-8 |
Table 4
For the following exercises, use each pair of functions to find and
72.
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75.
For the following exercises, use the functions and to evaluate or find the composite function as indicated.
78.
Extensions
For the following exercises, use and
80.
Find and Compare the two answers.
81.
Find and
82.
What is the domain of
83.
What is the domain of
84.
Let
- ⓐ Find
- ⓑ Is for any function the same result as the answer to part (a) for any function? Explain.
For the following exercises, let and
85.
True or False:
86.
True or False:
For the following exercises, find the composition when for all and
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89.
Real-World Applications
90.
The function gives the number of items that will be demanded when the price is The production cost is the cost of producing items. To determine the cost of production when the price is $6, you would do which of the following?
- ⓐ Evaluate
- ⓑ Evaluate
- ⓒ Solve
- ⓓ Solve
91.
The function gives the pain level on a scale of 0 to 10 experienced by a patient with milligrams of a pain-reducing drug in her system. The milligrams of the drug in the patient’s system after minutes is modeled by Which of the following would you do in order to determine when the patient will be at a pain level of 4?
- ⓐ Evaluate
- ⓑ Evaluate
- ⓒ Solve
- ⓓ Solve
92.
A store offers customers a 30% discount on the price of selected items. Then, the store takes off an additional 15% at the cash register. Write a price function that computes the final price of the item in terms of the original price (Hint: Use function composition to find your answer.)
93.
A rain drop hitting a lake makes a circular ripple. If the radius, in inches, grows as a function of time in minutes according to find the area of the ripple as a function of time. Find the area of the ripple at
94.
A forest fire leaves behind an area of grass burned in an expanding circular pattern. If the radius of the circle of burning grass is increasing with time according to the formula express the area burned as a function of time, (minutes).
95.
Use the function you found in the previous exercise to find the total area burned after 5 minutes.
96.
The radius in inches, of a spherical balloon is related to the volume, by Air is pumped into the balloon, so the volume after seconds is given by
- ⓐ Find the composite function
- ⓑ Find the exact time when the radius reaches 10 inches.
97.
The number of bacteria in a refrigerated food product is given by where is the temperature of the food. When the food is removed from the refrigerator, the temperature is given by where is the time in hours.
- ⓐ Find the composite function
- ⓑ Find the time (round to two decimal places) when the bacteria count reaches 6752.