Editing Function (mathematics) (section)

=== Function composition ===
{{Main|Function composition}}

Given two functions <math>f: X\to Y</math> and <math>g: Y\to Z</math> such that the domain of {{mvar|g}} is the codomain of {{mvar|f}}, their ''composition'' is the function <math>g \circ f: X \rightarrow Z</math> defined by

<math display="block">(g \circ f)(x) = g(f(x)).</math>

That is, the value of <math>g \circ f</math> is obtained by first applying {{math|''f''}} to {{math|''x''}} to obtain {{math|1=''y'' = ''f''(''x'')}} and then applying {{math|''g''}} to the result {{mvar|y}} to obtain {{math|1=''g''(''y'') = ''g''(''f''(''x''))}}. In this notation, the function that is applied first is always written on the right.

The composition <math>g\circ f</math> is an [[operation (mathematics)|operation]] on functions that is defined only if the codomain of the first function is the domain of the second one. Even when both <math>g \circ f</math> and <math>f \circ g</math> satisfy these conditions, the composition is not necessarily [[commutative property|commutative]], that is, the functions <math>g \circ f</math> and <math> f \circ g</math> need not be equal, but may deliver different values for the same argument. For example, let {{math|1=''f''(''x'') = ''x''<sup>2</sup>}} and {{math|1=''g''(''x'') = ''x'' + 1}}, then <math>g(f(x))=x^2+1</math> and <math> f(g(x)) = (x+1)^2</math> agree just for <math>x=0.</math>

The function composition is [[associative property|associative]] in the sense that, if one of <math>(h\circ g)\circ f</math> and <math>h\circ (g\circ f)</math> is defined, then the other is also defined, and they are equal, that is, <math>(h\circ g)\circ f = h\circ (g\circ f).</math> Therefore, it is usual to just write <math>h\circ g\circ f.</math>

The [[identity function]]s <math>\operatorname{id}_X</math> and <math>\operatorname{id}_Y</math> are respectively a [[right identity]] and a [[left identity]] for functions from {{mvar|X}} to {{mvar|Y}}. That is, if {{mvar|f}} is a function with domain {{mvar|X}}, and codomain {{mvar|Y}}, one has
<math>f\circ \operatorname{id}_X = \operatorname{id}_Y \circ f = f.</math>

<gallery widths="250" heights="300">
File:Function machine5.svg|A composite function ''g''(''f''(''x'')) can be visualized as the combination of two "machines".
File:Example for a composition of two functions.svg|A simple example of a function composition
File:Compfun.svg|Another composition. In this example, {{math|1=(''g'' ∘ ''f'' )(c) = #}}.
</gallery>