In mathematics, a composite function, formed by the composition of one function on another, represents the application of the former to the result of the application of the latter to the argument of the composite. The functions f: X → Y and g: Y → Z can be composed by first applying f to an argument x and then applying g to the result. Thus one obtains a function g o f: X → Z defined by (g o f)(x) = g(f(x)) for all x in X. The notation g o f is read as "g circle f" or "g composed with f".

gof, the composition of f and g

As an example, suppose that an airplane's height at time t is given by the function h(t) and that the oxygen concentration at height x is given by the function c(x). Then (c o h)(t) describes the oxygen concentration around the plane at time t.

In the mid-20th century, some mathematicians decided that writing "g of" to mean "first apply f, then apply g" was too confusing and decided to change notations. They wrote "xf" for "f(x)" and "xfg" for "g(f(x))". However, this movement never caught on, and nowadays this notation is found only in old books.

The functions g and f are commutative if g o f = f o g.

Derivatives of compositions involving differentiable functions can be found using the chain rule. See also Fa di Bruno's formula.

Functional powers

If Y⊂X then f may compose with itself; this is sometimes denoted f ². Thus:

(f o f)(x) = f(f(x)) = f ²(x)

(f  o f  o f)(x) = f(f(f(x))) = f ³(x)

The functional powers f of ⁿ = f ⁿ o f = f ⁿ⁺¹ for natural n follow immediately.

Do not confuse it with the notation commonly seen in trigonometry in which, for historical reasons, this superscript notation represents standard exponentiation when used with trigonometric functions: sin²(x) = sin(x) sin(x).

Nevertheless, an extension of this notation using negative exponents applies to all functions, including trigonometric ones:

f ^-1(x) is the inverse function of f

In some cases, an expression for f in g(x) = f ^r(x) can be derived from the rule for g given non-integer values of r. This is called fractional iteration.

Iterated functions occur naturally in the study of fractals and dynamical systems.

See also

• combinator
• lambda calculus
• higher-order function