Function composition is a fundamental concept in mathematics, particularly in algebra and calculus. In simple terms, it means applying one function to the result of another function.

1. Math30-1 1
10.3 Composite Functions
( ) 6
g x x
  ( ) 5
h x x
 
( ) 4
f x x

( ( ))
f g x 
( ) 4
f 
x x
( )
g x  
6
x 
4
 x  
4 6

4 24
x
 
How would
you define
composite
functions?

2. Math30-1 2
When two functions are applied in succession,
the resulting function is called the composite
of the two given functions.
Composition of Two Functions
(g o f)(x) = g(f(x))
The range of f(x) becomes the
domain of g(x).
Evaluate f(x) then use the answer as
the input for g(x).
Notice: The function is evaluated
from the inside to the outside.
x is the input into the f function to get y
y is the input into the g function to get z

3. Math30-1 3
( )
y f x

( )
y g f x
 
 
( )
y g f x

( )
y f g x
 
 
( )
y f g x

10.3 Function
Composition

4. Math30-1 4
( ) 6
g x x
  ( ) 5
h x x
 
( ) 4
f x x

( ( ))
h f x (4 )
h x

 
4 5
x
 
4 5
x
  5
,
4
x 

5. Math30-1 5
Given h(x) = 4x + 3, determine the following:
b) (h o h)(-3)
(h o h)(-3) = h(h(-3))
= h(4(-3) + 3)
= h(-9)
h(-9) = 4(-9) + 3
= -33
a) (h o h)(x)
(h o h)(x) = h(h(x))
= h(4(x) + 3)
= h(4x + 3)
h(4x + 3) = 4(4x + 3) + 3
= 16x + 12 + 3
= 16x + 15
Note: (h o h)(x) ≠ (hh)(x)
(hh)(x) = h(x) x h(x)
Evaluating a Composition of a Functions

6. Math30-1 6
Given ( ) 1
f x x
 
2
( ) 3
g x x
  ( ) 2 5
h x x
 
Determine an expression in simplest form for
( ( ))
h f x  
2 1 5
x
   , 1
x  Why are there restrictions?
  
( )
k x h g f x
  
 
 
( ) 1
k x h g x
 
 
 
2
( ) 1 3
k x h x
  
 
( ) 2
k x h x
 
 
( ) 2 2 5
k x x
  
( ) 2 1
k x x
  , 1
x 

7. Math30-1 7
Using the function
2
4
( )
2
x
f x
x



determine the value of  
 
4
f f
 
 
2
4 4
4
4 2
f f f
 

  

 
 
   
4 10
f f f

 
 
2
10 4
4
10 2
f f



 
 
4 13
f f 

8. Math30-1 8
5
( ) log
f x x

Determine the expressions for h(x)= f(g(x)) and k(x) = g(f(x))
( ) ( ( ))
h x f g x

( ) 5x
g x 
 
5
( ) log 5x
h x 
( )
h x x

( ) ( ( ))
k x g f x

5
log
( ) 5 x
k x 
( )
k x x

The situation when f(g(x)) = g(f(x)) = x indicates that the original
functions are inverses.

9. Math30-1 9
Use the graph to determine
the value of
 
 
4
g f
( )
y f x

( )
y g x

 (3)
f g
 (3) (3)
f g
 
0 ( 12)
  
12

 
2
g

12


10. Math30-1 10
Assignment
Page 507
1a,c, 2a,c, 3c, 4a,d,e, 5b,c, 6, 7, 8, 11, 13, 14, 15, 17
C1, C2