Worked examples on how to calculate the composite of functions as well as how to evaluate the composition of functions.

1. COMPOSITION OF
FUNCTIONS

2. OBJECTIVES:
• Define composition of functions.
• Perform composition of functions.
• Evaluate functional problems using composition of functions.

3. Composition of Functions
• Operation of function that must have two functions,
namely 𝒇(𝒙) and 𝒈 𝒙 ; and then perform the indicated
operation to produce the result.
• Also defined as, “applying a function to another
function”.

5. Example 1
𝑓 𝑥 = 3𝑥 − 1 and 𝑔 𝑥 = 𝑥 + 4
Find 𝑓 ∘ 𝑔 𝑥 .
• Insert the 𝒈(𝒙) function into the 𝒇 𝒙 function.
𝒇 ∘ 𝒈 𝒙 = 𝟑 𝒙 + 𝟒 − 𝟏
= 𝟑𝒙 + 𝟏𝟐 − 𝟏
= 𝟑𝒙 + 𝟏𝟏

6. Find an expression for 𝑓 ∘ 𝑔 𝑥 for the
following:
a) 𝒇 𝒙 = 𝒙𝟐 − 𝟔 𝒂𝒏𝒅 𝒈(𝒙) = 𝒙 + 𝟒
b) 𝒇 𝒙 = 𝒙 + 𝟑 𝒂𝒏𝒅 𝒈 𝒙 = 𝒙 − 𝟗
c) 𝒇 𝒙 = 𝟐𝒙 + 𝟒 𝒂𝒏𝒅 𝒈 𝒙 = 𝟖 − 𝟑𝒙
d) 𝒇 𝒙 =
𝟏
𝟐
𝒙 − 𝟒 𝒂𝒏𝒅 𝒈 𝒙 = 𝒙 + 𝟑

7. Check your answers here:
a) 𝑓 𝑥 = 𝑥2
− 6 𝑎𝑛𝑑 𝑔 𝑥 = 𝑥 + 4
𝑓 ∘ 𝑔 𝑥 = 𝑥 + 4 2 − 6
= 𝑥2 + 8𝑥 + 16 − 6
= 𝑥2 + 8𝑥 + 10
b) 𝑓 𝑥 = 𝑥 + 3 𝑎𝑛𝑑 𝑔 𝑥 = 𝑥 − 9
𝑓 ∘ 𝑔 𝑥 = 𝑥 − 9 + 3
= 𝑥 − 6
c) 𝑓 𝑥 = 2𝑥 + 4 𝑎𝑛𝑑 𝑔 𝑥 = 8 − 3𝑥
𝑓 ∘ 𝑔 𝑥 = 2 8 − 3𝑥 + 4
= 16 − 6𝑥 + 4
= 20 − 6𝑥
d) 𝑓 𝑥 =
1
2
𝑥 − 4 𝑎𝑛𝑑 𝑔 𝑥 = 𝑥 + 3
𝑓 ∘ 𝑔 𝑥 =
1
2
𝑥 + 3 − 4
=
1
2
𝑥 +
3
2
− 4
=
1
2
𝑥 −
5
2

8. Example 2
𝑓 𝑥 =
2
𝑥+3
and 𝑔 𝑥 =
−3𝑥−2
𝑥
. Find 𝑓 ∘ 𝑔 𝑥 .
• Insert the 𝒈(𝒙) function into the 𝒇 𝒙 function.
𝒇 ∘ 𝒈 𝒙 =
𝟐
−𝟑𝒙 − 𝟐
𝒙
+ 𝟑
=
𝟐
−𝟑𝒙 − 𝟐 + 𝟑𝒙
𝒙
=
𝟐
−
𝟐
𝒙
= 𝟐 ÷ −
𝟐
𝒙
= 𝟐 × −
𝒙
𝟐
= −𝒙

9. Evaluating Composite Functions
• Given that 𝑓 𝑥 = 4𝑥 + 3 and 𝑔 𝑥 = 𝑥 − 2, find 𝑓 𝑔 5 .
𝑓 𝑔 𝑥 = 4 𝑥 − 2 + 3
= 4𝑥 − 8 + 3
= 4𝑥 − 5
𝑓 𝑔 5 = 4 5 − 5
= 20 − 5
= 𝟏𝟓
Example 1

10. Evaluating Composite Functions
• Given that 𝑓 𝑥 = 6𝑥 − 4 and 𝑔 𝑥 = 𝑥 − 8, find 𝑓 𝑔 9 .
𝑓 𝑔 𝑥 = 6 𝑥 − 8 − 4 OR 𝑓 𝑔 5 = 6 𝑔 9 − 4
= 6𝑥 − 48 − 4 = 6 9 − 8 − 4
= 6𝑥 − 52 = 6 1 − 4
= 6 − 4 = 𝟐
𝑓 𝑔 9 = 6 9 − 52
= 54 − 52
= 𝟐
Example 2

11. EXERCISE!
Evaluate the following composite functions.
• 𝑓 𝑥 = 𝑥2 + 7
𝑔 𝑥 = 𝑥 − 3
Find 𝑓(𝑔 3 )
#1
• 𝑓 𝑥 = 𝑥 + 3
𝑔 𝑥 = 𝑥 − 5
Find 𝑔 ∘ 𝑓(2)
#2 • 𝑓 𝑥 = 7𝑥 + 4
𝑔 𝑥 = 2𝑥 − 4
Find 𝑔2(𝑥).
#3

12. Check answers!
#1
𝑓 𝑥 = 𝑥2
+ 7
𝑔 𝑥 = 𝑥 − 3
𝑔 3 = 3 − 3 = 0
∴ 𝑓 𝑔 3 = 𝑓 0
= 02 + 7
= 7
#2
𝑓 𝑥 = 𝑥 + 3
𝑔 𝑥 = 𝑥 − 5
𝑓 2 = 2 + 3 = 5

13. Check answers!
#3
𝑓 𝑥 = 7𝑥 + 4
𝑔 𝑥 = 2𝑥 − 4
Find 𝑔2
(𝑥).
𝑔2
(𝑥) is simply 𝑔 𝑔 𝑥 .
Therefore
𝑔2
𝑥 = 2 𝑥 − 4 − 4
= 2𝑥 − 8 − 4
= 𝟐𝒙 − 𝟏𝟐

14. The end.