Function composition

In mathematics, function composition is a way to make a new function from two other functions.

If we let f be a function from X to Y and g be a function from Y to Z then we say that g composed with f is written as g ∘ f a function from X to Z (notice how it is usually written in the opposite way to how people would it expect it to be as we will explain below).

The value of f given the input x is written as f(x). The value of g ∘ f given the input x is written (g ∘ f)(x) and is defined as g(f(x)) (which means our way of writing g composed with f makes sense).

Here is another 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 would be written as:

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

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

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

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

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]

Properties

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

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

Function composition is in general not commutative however, which means:

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

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