What Is a Composite Function?

What Is a Composite Function?

A composite function is a function of a function.

A composite function combines two or more functions so that the output of one function becomes the input of another.

Imagine a function f that relates an input x to an output f(x). The output is passed into another function g, which relates it to an output gf(x).

The composite function gf relates the input x straight to the output gf(x).

The composite function is denoted gf(x), (g ∘ f)(x) or g(f(x)).

A Real Example of a Composite Function

It is easier to understand composite functions with an example.

f(x) = 2x and g(x) = x + 1

Consider two functions:

f(x) = 2x

g(x) = x + 1

  • The function f(x) = 2x takes each input and doubles it.

  • The function g(x) = x + 1 takes each input and adds 1 to it.

The mapping diagrams below show these functions.

The Composite Function gf(x)

Now let's consider the composite function of f(x) = 2x and g(x) = x + 1.

  • The function f relates an input x to an output 2x (it doubles the input).

  • The output 2x is the input of the function g, which relates it to an output 2x + 1 (it adds 1 to the input).

The mapping diagram below shows this.

The composite of f(x) = 2x and g(x) = x + 1 is:

The Order of Composite Functions Matter

In the example above, the output of f(x) became the input of g(x). The composite function was gf(x) = 2x + 1.

Consider what happens if the output of g(x) became the input of f(x).

  • The function g relates an input x to an output x + 1 (it adds 1 to the input).

  • The output x + 1 is the input of the function f, which relates it to an output 2(x + 1) (it doubles the input).

The mapping diagram below shows this.

The composite of g(x) = x + 1 and f(x) = 2x is:

fg(x) is not the same as gf(x). The order of the functions matter.

gf(x)fg(x)

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See Also

What is a function? What is an inverse function?