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

What are Composite Functions -
Professor Edward Burger explains composite functions

Components of composite functions -
Professor Edward Burger explains components of composite functions

Finding functions that form a given composite -
Professor Edward Burger explains finding functions that form a given composite.

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