If we are given two functions, we can create another function by composing one function into the other. The steps required to perform this operation are similar to when any function is solved for any given value. Such functions are called composite functions.
What is a Composite Function?
A composite function is generally a function that is written inside another function. Composition of a function is done by substituting one function into another function.
For example, f [g (x)] is the composite function of f (x) and g (x). The composite function f [g (x)] is read as “f of g of x”. The function g (x) is called an inner function and the function f (x) is called an outer function. Hence, we can also read f [g (x)] as “the function g is the inner function of the outer function f”.
How to Solve Composite Functions?
Solving a composite function means, finding the composition of two functions. We use a small circle (∘) for the composition of a function. Here are the steps on how to solve a composite function:
Rewrite the composition in a different form.
For example
(f ∘ g) (x) = f [g (x)]
(f ∘ g) (x²) = f [g (x²)]
Substitute the variable x that is in the outside function with the inside function.
Simplify the function.
Note: The order in the composition of a function is important because (f ∘ g) (x) is NOT the same as (g ∘ f) (x).
Let’s look at the following problems:
Example 1
Given the functions f (x) = x2 + 6 and g (x) = 2x – 1, find (f ∘ g) (x).
Solution
Substitute g (x) with 2x – 1 in the function f(x) = x2 + 6.
(f ∘ g) (x) = (2x – 1)2 + 6 = (2x – 1) (2x – 1) + 6
Apply FOIL
= 4x2 – 4x + 1 + 6
= 4x2 – 4x + 7
Example 2
Given the functions g (x) = 2x – 1 and f (x) = x2 + 6, find (g ∘ f) (x).
Solution
Substitute x with x2 + 6 in the function g (x) = 2x – 1
(g ∘ f) (x) = 2(x2 + 6) – 1
Use the distributive property to remove the parentheses.
= 2x2 + 12 – 1
= 2x2 + 11
Example 3
Given f (x) = 2x + 3, find (f ∘ f) (x).
Solution
(f ∘ f) (x) = f[f(x)]
= 2(2x + 3) + 3
= 4x + 9