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.

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)

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:

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