1) A composite function is formed by combining two functions, where one function is substituted into the other. 2) Notations like fg(x) indicate that function g(x) is substituted into function f(x). 3) Composite functions are non-commutative, meaning the order of the functions matters - fg(x) may not equal gf(x).

1. Composite Functions
Prepared by: Alona Hall

2. At the end of this lesson, students should be
able to:
1. Describe the process involved in deriving a composite function
2. Identify the notations related to composite functions
3. Derive composite functions given two functions
4. Explain what is meant by the non-commutativity of composite
functions

3. Definition of the term ‘composite function’
According to the Oxford Dictionary, the term ‘composite’ means
‘something made by putting together different parts or materials.’
Thus, a composite function is a function that results when two or more
functions are put together.
This course requires that you learn to derive a composite function of
not more than two functions.

4. The Process
In order to combine two functions in order to obtain a resulting
function, one function is simply substituted into the other.
Good foundational knowledge on the concept of substitution will be
very useful for this topic

5. Notations used for Composite Functions
There are particular notations used to denote composite functions. The
notation gives an understanding as to which function should be
substituted into the other. For example,
• fg (x): means that the function g(x) is to be substituted into the
function f(x)
• ff(x): means that the function f(x) is to be substituted into itself (i.e
f(x)). Sometimes the notation 𝑓2 is used to mean the same
thing.

6. Examples
Given the functions 𝑓 𝑥 = 𝑥3 − 2, 𝑔 𝑥 = 3𝑥 − 5, 𝑑𝑒𝑡𝑒𝑟𝑚𝑖𝑛𝑒:
a) 𝑓𝑔 𝑥
b) 𝑔𝑓 𝑥
c) ff(x)

7. Solution
a) 𝑓𝑔 𝑥
. Substitute 𝑔 𝑥 𝑖𝑛𝑡𝑜 𝑓(𝑥): Now 𝑓 𝑥 = 𝑥3 − 2, 𝑔 𝑥 = 3𝑥 − 5
So 𝑓𝑔 𝑥 = 𝑓[𝑔(𝑥)]
i.e. [𝑔(𝑥)]3 −2 … … 𝑁𝐵: 𝑤𝑒 𝑙𝑜𝑜𝑘 𝑓𝑜𝑟 𝑡ℎ𝑒 𝑥 𝑡𝑒𝑟𝑚/𝑠 in f(x) and
substitute g(x) in its/their place.
Thus, we get: (3x – 5)3
−2
NOTE: The answer can be left like this without expanding the brackets.

8. Solution
b) 𝑔𝑓 𝑥
• Substitute f(x) into g(x). Recall 𝑓 𝑥 = 𝑥3 − 2, 𝑔 𝑥 = 3𝑥 − 5
𝑆𝑜 𝑔𝑓 𝑥 = 𝑔[𝑓(𝑥)]
𝑖. 𝑒. 𝑔 𝑓 𝑥 = 3 𝑓 𝑥 − 5
𝑇ℎ𝑢𝑠, 𝑤𝑒 𝑔𝑒𝑡: 3 𝑥3
− 5 − 5. In this case we can remove the brackets since the terms in the bracket are raised to the
power of 1.
𝑔 𝑓 𝑥 = 3𝑥3
− 15 − 5
= 3𝑥3
− 20
Note: 𝑓𝑔 𝑥 ≠ 𝑔𝑓 𝑥 .
𝐼𝑛 𝑜𝑡ℎ𝑒𝑟 𝑤𝑜𝑟𝑑𝑠, 𝑡ℎ𝑒 𝑟𝑒𝑠𝑢𝑙𝑡𝑠 𝑎𝑟𝑒 𝑛𝑜𝑡 𝑡ℎ𝑒 𝑠𝑎𝑚𝑒 𝑖𝑓 𝑡ℎ𝑒 𝑜𝑟𝑑𝑒𝑟 𝑖𝑛 𝑤ℎ𝑖𝑐ℎ 𝑡ℎ𝑒 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛𝑠 𝑎𝑟𝑒 𝑠𝑢𝑏𝑠𝑡𝑖𝑡𝑢𝑡𝑒𝑑 𝑐ℎ𝑎𝑛𝑔𝑒𝑠.
We say that composite functions are NON-COMMUTATIVE.

9. Solutions
c) 𝑓𝑓 𝑥 or 𝑓2
• Substitute f(x) into f(x). Recall 𝑓 𝑥 = 𝑥3 − 2,
𝑆𝑜 𝑓𝑓 𝑥 = 𝑓[𝑓(𝑥)]
𝑖. 𝑒. 𝑓 𝑓 𝑥 = 𝑓 𝑥 3 − 2
𝑇ℎ𝑢𝑠, 𝑤𝑒 𝑔𝑒𝑡: 𝑥3
− 2 3
− 2. Again, we need not expand the brackets.
𝑓 𝑓 𝑥 = 𝑥3 − 2 3 − 2