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Name: Block: Date: Pre-Calculus 11
Chapter 5A Functions
Lesson #2 Composition of Functions
Warm it up
Consider: 𝑓(𝑥) = 2√x − 1
𝑔(𝑥) = 5√x + 3
Calculate the following and state the domain and range
1. (f + g)(x)
2. (f ─ g)(x)
3. (f · g)(x)
4. ( f
g )(x)
Investigation
Consider: 𝑓(𝑥) = 5𝑥
𝑔(𝑥) = 𝑥2 + 3
1. Complete the following mapping by putting output values of previous functions to become
the input values of the subsequent functions.
2. Using the following mapping, what will be the algebraic equation of the last mapping?
Explain.
3. Does it make a difference when we change the order of the functions for our mapping?
Explain.
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Name: Block: Date: Pre-Calculus 11
Composition of Functions
e.g.
Consider: 𝑓(𝑥) = √x
𝑔(𝑥) = 2𝑥 − 3
Example #1
Consider: 𝑓(𝑥) = 𝑥2
𝑔(𝑥) = 𝑥 − 1
Calculate
a) (f ○ g) (x) (f ○ g) (-5)
b) (g ○ f) (x) (g ○ f) (2)
c) (f ○ f) (x) (f ○ f) (-3)
d) (g ○ g) (x) (g ○ g) (2)
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Name: Block: Date: Pre-Calculus 11
Example #2
Consider: 𝑓(𝑥) = 𝑥2 + 1 𝑔(𝑥) = −2𝑥
Calculate, and draw the graph of each of these composite functions
a) (f ○ g) (x)
b) (g ○ f) (x)
c) (f ○ f) (x)
d) (g ○ g) (x)
Example #3
Consider: 𝑓(𝑥) = √2x − 5
𝑔(𝑥) = 3𝑥 − 2
Calculate, and state the domain and range for each of these composite functions
a) (f ○ g) (x) Domain: Range:
b) (g ○ f) (x) Domain: Range:
c) (f ○ f) (x) Domain: Range:
d) (g ○ g) (x) Domain: Range:
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Name: Block: Date: Pre-Calculus 11
Example #4
If ℎ(𝑥) = 𝑓(𝑔(𝑥)), determine 𝑓(𝑥) 𝑎𝑛𝑑 𝑔(𝑥), where 𝑓(𝑥) ≠ 𝑔(𝑥).
a) ℎ(𝑥) = (𝑥 − 3)2 + (𝑥 − 3) + 1
b) ℎ(𝑥) = √x2 − 1
c) ℎ(𝑥) = 𝑥
Homework
1. Pg. 256-257 #1, 6, 9a, 10, 12, 15, 22, 25, 29, 32, 35, 37 2. Pg. 257-259 #40, 44, 48, 50, 54, 60, 63, 65, 74, 85