Composite functions

A composite function is a function created when one function is used as the input value for another function. Essentially, the output of the inner function (the function used as the input value) becomes the input of the outer function (the resulting value).

For the functions f(x) and g(x), when g(x) is used as the input of f(x), the composite function is written as:

(f ∘ g)(x)

The symbol denotes a composite function - it looks similar to the multiplication symbol, , but does not mean the same thing. (f ∘ g)(x) is the same thing as f(g(x)).

(f ∘ g)(x) is not the same thing as (g ∘ f)(x). (g ∘ f)(x) is the same thing as g(f(x)), which will often be different than f(g(x)).

You can use composite functions to check if two functions are inverses of each other because they will follow the rule:

(f ∘ g)(x) = (g ∘ f)(x) = x

You can find the composite of two functions by replacing every x in the outer function with the equation for the inner function (the input).

Example

Given:  f(x) = 4x2 + 3;  g(x) = 2x + 1

(f ∘ g)(x) = f(g(x)) = 4(2x + 1)2 + 3
  = 4(4x2 + 4x + 1) + 3
  = 16x2 + 16x + 7
(f ∘ g)(x) = 16x2 + 16x + 7

(g ∘ f)(x) = g(f(x)) = 2(4x2 + 3) + 1
  = 8x2 + 6 + 1
(g ∘ f)(x) = 8x2 + 7

Just like with inverse functions, you need to apply domain restrictions as necessary to composite functions. The composite of two functions f(x) and g(x) must abide by the domain restrictions of f(x) and g(x). In the example above, both functions had domains of all real numbers, so their composite functions did not have any domain restrictions either.

Let's look at an example where domain restrictions apply.

Example

Given: 

Although g(x) = x2 has a domain of all real numbers, has a domain of [0, ∞). Therefore, the composite function (f ∘ g)(x) and (g ∘ f)(x) both have a domain restriction of [0, ∞).


See also inverse functions.