Function composition

In mathematics, the composition operator

∘

{\displaystyle \circ }

takes two functions,

f

{\displaystyle f}

and

g

{\displaystyle g}

, and returns a new function

f

∘

g

{\displaystyle f\circ g}

. When the composite function

f

∘

g

{\displaystyle f\circ g}

(pronounced "

f

{\displaystyle f}

of

g

{\displaystyle g}

") is evaluated at an input

x

{\displaystyle x}

, the result is

(

f

∘

g

)

(

x

)

=

f

(

g

(

x

)

)

{\displaystyle (f\circ g)(x)=f(g(x))}

. That is, the function

f

{\displaystyle f}

is applied after applying

g

{\displaystyle g}

to

x

{\displaystyle x}

.

The composition of functions is a special case of the composition of relations, sometimes also denoted by

∘

{\displaystyle \circ }

. As a result, all properties of composition of relations are true of composition of functions, such as associativity.

Examples

Composition of functions on a finite set: If f = {(1, 1), (2, 3), (3, 1), (4, 2)}, and g = {(1, 2), (2, 3), (3, 1), (4, 2)}, then g ∘ f = {(1, 2), (2, 1), (3, 2), (4, 3)}, as shown in the figure.

Composition of functions on an infinite set: If f: R → R (where R is the set of all real numbers) is given by f(x) = 2x + 4 and g: R → R is given by g(x) = x3, then:

If an airplane's altitude at time t is a(t), and the air pressure at altitude x is p(x), then (p ∘ a)(t) is the pressure around the plane at time t.

Function defined on finite sets which change the order of their elements such as permutations can be composed on the same set, this being composition of permutations.

Properties

The composition of functions is always associative—a property inherited from the composition of relations. That is, if f, g, and h are composable, then f ∘ (g ∘ h) = (f ∘ g) ∘ h.