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GP1-‐PAP-‐S3-‐HW6-‐ Answers
Skill 3e Day 1: Compositions of Functions
1. If 𝑓 𝑥 = 𝑥! − 4𝑥! + 3𝑥 − 9 and 𝑔 𝑥 = 𝑥! + 5𝑥! − 6𝑥 − 1, find
A) 𝑓 + 𝑔 𝑥 = 2𝑥! + 𝑥! − 3𝑥 − 10
B) 𝑓 − 𝑔 𝑥 = −9𝑥! + 9𝑥 − 8
2. If 𝑓 𝑥 = 8𝑥! + 3𝑥 − 9 and 𝑔 𝑥 = 𝑥! + 3𝑥 − 16, find
A) 𝑓 + 𝑔 𝑥 = 𝑥! + 8𝑥! + 6𝑥 − 25
B) 𝑓 − 𝑔 𝑥 = −𝑥! + 8𝑥! + 7
C) (𝑔 + 𝑓)(𝑥) = 𝑥! + 8𝑥! + 6𝑥 − 25
D) 𝑔 − 𝑓 𝑥 = 𝑥! − 8𝑥! − 7
E) What is the domain of each of these?
All of these are polynomial functions. All polynomial functions have a domain of all real numbers.
𝐷 = −∞,∞
3. If 𝑓 𝑥 = !!!!
and 𝑔 𝑥 = !!!!
, find
A) 𝑓 + 𝑔 𝑥 = !!!!!!"!!! !!!
B) 𝑓 − 𝑔 𝑥 = !!!!!!!!"!!! !!!
C) (𝑔 + 𝑓)(𝑥) = !!!!!!"!!! !!!
D) (𝑔 − 𝑓)(𝑥) = !!!!!!!"!!! !!!
E) What is the domain of each of these?
The denominator determines the domain of rational functions. All of these rational functions have the same denominator. All of these functions have the same domain.
𝐷 = −∞,−4 ∪ −4, 4 ∪ 4,∞
4. If 𝑓 𝑥 = 5 − 𝑥 and 𝑔 𝑥 = 𝑥 + 6, What is the domain of (𝑓 + 𝑔)(𝑥)?
The function 𝑓 has a domain; 𝐷 = (−∞, 5].
The function 𝑔 has a domain; 𝐷 = [−6,∞).
The function 𝑓 + 𝑔 has a domain; 𝐷 = −6, 5
5. If 𝑓 𝑥 = !!!!!!!"
and 𝑔 𝑥 = !!!"!!!
, What is the domain of (𝑓 − 𝑔)(𝑥)?
The function 𝑓 has a domain; 𝐷 = −∞, 4 ∪ 4,∞ .
The function 𝑔 has a domain; 𝐷 = (−∞, 8].
The function 𝑓 − 𝑔 has a domain; 𝐷 = −∞, 4 ∪ (4, 8]
6. If 𝑓 𝑥 = 4 − 𝑥! and 𝑔 𝑥 = − !!!!!
, What is the domain of (𝑓 + 𝑔)(𝑥)?
The function 𝑓 has a domain; 𝐷 = −2, 2 .
The function 𝑔 has a domain; 𝐷 = −∞,−1 ∪ −1, 1 ∪ 1,∞ .
The function 𝑓 − 𝑔 has a domain; 𝐷 = [−2,−1) ∪ (−1, 1) ∪ (1, 2]
𝑓 𝑥 and 𝑔(𝑥) are graphed below. If ℎ 𝑥 = 𝑓 𝑥 + 𝑔(𝑥), sketch a graph of ℎ(𝑥)
7.
𝑥 𝑓(𝑥) 𝑔(𝑥) ℎ(𝑥) −8 9 −2 7 −6 8 −1 7 −4 7 0 7 0 5 2 7
8.
𝑥 𝑓(𝑥) 𝑔(𝑥) ℎ(𝑥) −3 9 −5 4 −2 4 −4 0 −1 1 −3 −2 0 0 −2 −2 1 1 −1 0 2 4 0 4
8.
𝑓 𝑥 and 𝑔(𝑥) are graphed below. If ℎ 𝑥 = 𝑓 𝑥 − 𝑔(𝑥), sketch a graph of ℎ(𝑥) 9.
𝑥 𝑓(𝑥) 𝑔(𝑥) ℎ(𝑥) −8 9 −2 11 −6 8 −1 9 −4 7 0 7 0 5 2 3
10.
𝑥 𝑓(𝑥) 𝑔(𝑥) ℎ(𝑥) −3 9 4 5 −2 4 4 0 −1 1 4 −3 0 0 4 −4 1 1 4 −3 2 4 4 0 3 9 4 5
𝑓 𝑥 is graphed below. If ℎ 𝑥 = 𝑓 𝑥 + 𝑓 𝑥 = 2𝑓(𝑥), sketch a graph of ℎ(𝑥) 11.
𝑥 𝑓(𝑥) ℎ(𝑥) −8 6 12 −6 5 10 −4 4 8 4 0 0 6 −1 −2 9 −1 −2
12.
𝑥 𝑓(𝑥) ℎ(𝑥) −2 0 0 0 −3 −6 2 −4 −8 4 −3 −6 6 0 0
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