Composite Functions

In this lesson, we will look at the composition of functions.

The composition of f and g is written as (f ο g)(x). Another form of composition notation is f(g(x)) or fg(x), read as “f of g of x”.

In the composition (f ο g)(x), the domain of f becomes g(x).

Example:

Given f(x) = 3x + 2 and g(x) = x + 5, find

a) (f ο g)(x)
b) (g ο f)(x)

Solution:

a) (f ο g)(x)
= f(x + 5)
= 3(x + 5) + 2
= 3x + 15 + 2
= 3x + 17

b) (g o f)(x)
= g(3x + 2)
= 3x + 2 + 5
= 3x + 7

Example:

Given f(x) = x² + 6 and g(x) = 2x – 1, find

a) (f ο g)(x)
b) (g ο f)(x)

Solution:

a) (f ο g)(x)
= f(2x – 1)
= (2x – 1)² + 6
= 4x² – 4x + 2 + 6
= 4x² – 4x + 8

b) (g ο f)(x)
= g(x² + 6)
= 2(x² + 6) – 1
= 2x² + 12 – 1
= 2x² + 11

Videos

Composition of Functions

Composite functions
Finding a composition of two functions

Composite Functions
This lesson explains the concept of composite functions. An example is given demonstrating how to work algebraically with composite functions and another example involves an application that uses the composition of functions.

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