This lesson covers composite functions in the context of Cambridge IGCSE Mathematics. It explains how to find composite functions through substitution, using examples to demonstrate the process. Students are also given practice problems to reinforce their understanding of the topic.

1. Version 1
Lesson slides
Topic E2.13: Functions
Lesson 2: Composite functions
Cambridge IGCSE™
Mathematics 0580

2. Starter – substitution practice
Substitute into, and simplify, the following equations.
Substitute into
Substitute into
Substitute into
𝒚 =𝟏𝟎 𝒙 −𝟏
𝒚 =𝟗 𝒙𝟐
𝒚 =𝟏𝟐 𝒙 −𝟑

3. What is a composite function?
If something is a ‘composite’ then it is made up of two or more things.
A composite function is two or more functions being substituted into one another.

4. What is a composite function?
Say we have two functions: f (𝑥)=2 𝑥 +5 g( 𝑥 )= 𝑥2
We can now find the composite function
You start with the function closest to the input, and then work your way backwards.
Start by finding .
This is pronounced: ‘f’ of ‘g’ of 2
𝐟𝐠 (𝟐 )=𝐟 (𝟐𝟐
)
¿ 𝐟 ( 𝟒 )
𝐟 (𝟒)=𝟐 (𝟒)+𝟓
¿ 𝟖 +𝟓
¿ 𝟏𝟑
𝐟𝐠 (𝟐)=𝟏𝟑

5. Example 1
Find
𝐟𝐠 (𝟏)=𝐟 (𝐠(𝟏))
¿ 𝐟 (𝟐+ 𝟒 (𝟏))
¿ 𝐟 ( 𝟔 )
𝐟 (𝟔)=𝟔𝟐
+ 𝟔
¿ 𝟑𝟔 +𝟔
¿ 𝟒𝟐

6. Example 2
Find
𝐠𝐟 (𝟏)=𝐠(𝐟 (𝟏))
¿ 𝐠 (𝟏𝟐
+𝟔)
¿ 𝐠 ( 𝟕 )
𝐠(𝟕)=𝟐 +𝟒(𝟕)
¿ 𝟐+𝟐𝟖
¿ 𝟑𝟎

7. Your turn
1. Find
𝐟𝐠 (𝟒)=𝐟 (𝟑(𝟒))
¿ √𝟏𝟐+ 𝟒
¿ √ 𝟏𝟔
¿ 𝟒
2. Find
𝐠𝐡(−𝟒)=𝐠(𝟐(−𝟒)+𝟏)
¿ 𝐠 (− 𝟕)
¿ 𝟑( − 𝟕)
¿ − 𝟐𝟏
Non-Calculator

8. Your turn
1. Find Give the answer correct to 2 decimal places.
𝐟𝐡(𝟏𝟏)=𝐟(𝟐(𝟏𝟏)+𝟏)
¿ √𝟐𝟑+ 𝟕
¿ 𝟓 . 𝟐𝟎
Calculator

9. Finding the equation of composite functions
f (𝑥 )=2 𝑥 +1
Find
g ( 𝑥)=3 𝑥2
+ 6
No input this time. We substitute one function into another:
¿12 𝑥2
+12 𝑥+9
Notice how they are different. The order of the functions is important.
Find
𝐟𝐠 ( 𝒙)=𝒇 (𝟑 𝒙𝟐
+𝟔)
¿ 𝟐 (𝟑 𝒙
𝟐
+ 𝟔)+𝟏
¿ 𝟔 𝒙𝟐
+𝟏𝟐 +𝟏
¿ 𝟔 𝒙 𝟐
+𝟏𝟑
𝐟 ( 𝒙)=𝒈 (𝟐 𝒙+𝟏)
¿ 𝟑(𝟐 𝒙+ 𝟏)𝟐
+𝟔
¿ 𝟑(𝟒 𝒙𝟐
+𝟒 𝒙+𝟏)+𝟔
¿𝟏𝟐 𝒙𝟐
+𝟏𝟐 𝒙+𝟑+𝟔
¿ 𝟏𝟐 𝒙𝟐
+𝟏𝟐 𝒙+𝟗

10. Your turn
f (𝑥 )=3 𝑥 −1
1. Find
g ( 𝑥)=4 − 𝑥2
2. Find
𝐟𝐠(𝒙)= 𝒇 (𝟒− 𝒙𝟐
) 𝐠𝐟 (𝒙 )=𝒈(𝟑 𝒙 − 𝟏)
¿𝟒−(𝟗 𝒙𝟐
−𝟔 𝒙 +𝟏)

11. Your turn – challenge
, Find the value of
𝐟𝐠 (𝒃)=𝐟 ( 𝒃−𝟏)=−𝟏𝟒
(𝒃− 𝟏)𝟐
−𝟕(𝒃− 𝟏)− 𝟐=− 𝟏𝟒
𝒃𝟐
−𝟐 𝒃+𝟏−𝟕𝒃+𝟕−𝟐=−𝟏𝟒
𝒃𝟐
−𝟗 𝒃+𝟔=−𝟏𝟒
𝒃𝟐
−𝟗 𝒃+𝟐𝟎=𝟎
(𝒃−𝟒)(𝒃−𝟓)=𝟎
or

12. Now work through worksheet 2