Unit8
MHF 4U1
Lesson 8 – Composition of Functions
Consider the function f ( x)  2 x  5 .
An analogy for how this function works is to think of f (x) as a “machine”
with an input and an output. This particular machine takes the input,
doubles it, and then adds 5.
x
f (x )
2x  5
f (3)
2(3)  5  11
f (3) can be represented by:
3
Composition Notation
f  g (x) is known as “f composed with g at x”
It is the notation used for the following process:
1. x is used as the input to g (x ) .
2. The output from g (x ) is placed as the input to f (x) .
3. The final result is the output from f (x) .
Another way to write this process is: f  g ( x)  f ( g ( x)) .
To better understand this process, consider two functions: f ( x)  3x  1 and
g ( x)  x 2 .
f (g (2)) is the final result when 2 is used as the input to g (x ) and the output
from g (x ) is placed into f (x) .
g ( 2)
22  4
4
f ( 4)
3(4)  1  11
The proper notation for this process is:
f ( g (2))
 f (2 2 )
 f (4)
 3(4)  1
 11
Note that g  f (2) or g ( f (2)) is not necessarily the same as f  g (2) or f (g (2)) .
Unit8
MHF 4U1
For g ( f (2)) , we have to use 2 as an input into f (x) first, and then place the
result as an input into g (x ) .
2
f ( 2)
3(2)  1  5
5
g (5)
5 2  25
The proper notation for this process is:
g ( f (2))
 g (3(2)  1)
 g (5)
 52
 25
Example 1:
Given that f ( x)  2 x  3 and g ( x) 
x , determine whether ( f  g )( x)  ( g  f )( x)