Composite Functions

Show your work for each problem.

1.
If and , what is ?
1. a)4
2. b)9
3. c)5
4. d)6
Answer: 5
. First, . Then, .
2.
What does mean?
1. a)
2. b)
3. c)
4. d)
Answer:
means apply first, then apply to the result. This is function composition.
3.
If and , find .
Answer: 9
. Then . So .
4.
If and , find .
Answer: 5
. Then . So .
5.
If and , find .
Answer: 11
1. What does mean in terms of and ? f(g(3))
2. Calculate 6
3. Now calculate 11
6.
If and , what is ?
1. a)
2. b)
3. c)
4. d)
Answer:
. Note: and are equivalent, but is the direct answer.
7.
If and , find .
Answer: 17
. Then .
8.
If and , find .
Answer: 19
. Then .
9.
If and , find the formula for .
Answer: 2x^2 - 3
1. What is ? x^2
2. Substitute into . What is ? 2x^2 - 3
10.
If and , compare and . Are they equal?
Answer: no
1. Calculate . First, 8
2. Now 7
3. Calculate . First, 1
4. Now 4
5. Is ? (yes/no) no
11.
True or False: For all functions and , .
1. a)False
2. b)True
Answer: False
False. Composition is generally NOT commutative. For example, if and , then but .
12.
If and , find .
Answer: 4
. Then .
13.
If , , and , find .
Answer: 11
, , . So .
14.
If and , find as a simplified expression.
Answer: 3x + 1
1. . What is ? x + 2
2. Substitute into . What is ? 3(x + 2) - 5
3. Expand and simplify 3x + 1
15.
If , find functions and such that where .
Answer: 2x - 1
1. If and , what must be so that ? 2x - 1
16.
If and , what is ?
1. a)
2. b)
3. c)
4. d)
Answer:
. Then .