Function composition

In mathematics, function composition is a way of making a new function from two other functions through a chain-like process.

More specifically, given a function f from X to Y and a function g from Y to Z, then the function "g composed with f", written as g ∘ f, is a function from X to Z (notice how it is usually written in the opposite way to how people would expect it to be).

The value of f given the input x is written as f(x). The value of g ∘ f given the input x is written as (g ∘ f)(x), and is defined as g(f(x)).

As an example. let f be a function which doubles a number (multiplies it by 2), and let g be a function which subtracts 1 from a number. These two functions can be written as:

[math]\displaystyle{ f(x) = 2x }[/math]

[math]\displaystyle{ g(x) = x - 1 }[/math]

Here, g composed with f would be the function which doubles a number, and then subtracts 1 from it. That is:

[math]\displaystyle{ (g \circ f)(x) = 2x - 1 }[/math]

On the other hand, f composed with g would be the function which subtracts 1 from a number, and then doubles it:

[math]\displaystyle{ (f \circ g)(x) = 2(x-1) }[/math]

Composition of functions can also be generalized to binary relations, where it is sometimes represented using the same [math]\displaystyle{ \circ }[/math] symbol (as in [math]\displaystyle{ R \circ S }[/math]).

Properties

Function composition can be proven to be associative, which means that:

[math]\displaystyle{ f \circ (g \circ h) = (f \circ g) \circ h }[/math]

However, function composition is in general not commutative, which means that:

[math]\displaystyle{ f \circ g \neq g \circ f }[/math]

This can be also seen in the first example, where (g ∘ f)(2) = 2*2 - 1 = 3 and (f ∘ g)(2) = 2*(2-1) = 2.

Related pages

• Chain rule