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:

${\displaystyle f(x)=2x}$

${\displaystyle g(x)=x-1}$

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

${\displaystyle (g\circ f)(x)=2x-1}$

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

${\displaystyle (f\circ g)(x)=2(x-1)}$

Composition of functions can also be generalized to binary relations, where it is sometimes represented using the same ${\displaystyle \circ }$ symbol (as in ${\displaystyle R\circ S}$).^[1]

Function composition can be proven to be associative, which means that:^[2]

${\displaystyle f\circ (g\circ h)=(f\circ g)\circ h}$

However, function composition is in general not commutative, which means that:^[3]

${\displaystyle f\circ g\neq g\circ f}$

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.

• Chain rule