1. Composition of
Functions 1

2. Composition of Functions
When two functions are applied in succession,
the resulting function is called the composite
of the two given functions.
One function is inserted within the other in the place
of x, and then the resulting function is simplified. The new function is called the explicit function.
Symbols:
f(g(x)) can also be written as (f o g)(x)
g(f(x)) can also be written as (g o f)(x)

3. Determining a Composition of Two Functions
Given f(x) = 8x - 1 and g(x) = 3x2 + 4x, find the following:
a. (f o g)(x)
This means we sub g(x) into f(x).
(f o g)(x) = 8(3x2 + 4x) – 1
=24x2 + 32x – 1
b. (g o f)(x)
(g o f)(x) = 3(8x – 1 )2 + 4(8x – 1)
=3(8x – 1)(8x -1) + 4(8x – 1)
= 3(64x2 – 16x + 1) + 32x – 4
= 192x2 - 48x x – 4
= 192x2 - 16x - 1

4. Examples:
For each function below, determine possible functions f and g so that y =f(g(x)).
a. y = (x + 4)2 b.

5. Determine domain and range of the composite function y = f o g(x) if f(x) = and g(x) = x – 6.

6. Evaluating a Composition of Two Functions
Given f(x) = 8x - 1 and g(x) = 3x + 4, find the following:
b) (g o f)(2)
This tells you to find the
value of g(x) first, then use
this value for the function
f(x). Solve for g(2).
a) (f o g)(2)
(g o f)(x) = g(f(x))
(f o g)(x) = f(g(x))
f(x) = 8x - 1
f(2) = 8(2) - 1
= 15
g(x) = 3x + 4
g(2) = 3(2) + 4
= 10
With the value of g(2) = 10,
you now solve for f(10).
(g o f)(2) = g(f(2))
= g(15)
(f o g)(2) = f(g(2))
= f(10)
g(2) = 10
g(x) = 3x + 4
g(15) = 3(15) + 4
= 49
f(x) = 8x - 1
f(10) = 8(10) - 1
= 79
(f o g)(2) = 79
(g o f)(2) = 49 2

7. Evaluating a Composition of a Function With Itself
Given h(x) = 4x + 3, find the following:
a) (h o h)(-3)
b) (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
= 16x + 15
(h o h)(-3) = h(h(-3))
= h(4(-3) + 3)
= h(-9)
h(-9) = 4(-9) + 3
= -33
Note: (h o h)(x) ≠ (hh)(x)
(hh)(x) = h(x) x h(x)
5.2.3