Function composition

In mathematics, a composite function, or composition of one function on another, represents the result (value) of one function used as the argument (i.e., the "input") to another.

In the expression

the value of g is the parameter of f, and the function f is composed on g. An equivalent representation is

f.g is a function which is the composite function of f on g; read "f circle g" or "f composed with g".

Derivatives of compositions involving differentiable functions can always be found using the chain rule.

The composition of a function on itself, such as f.f, is customarily written f². (f.f)(x)=f(f(x))=f²(x). Likewise, (f.f.f)(x)=f(f(f(x)))=f³(x). By extension of this notation, f^-1(x) is the inverse function of f.

However, for historical reasons, this superscript notation does not mean the same thing for trigonometric functions unless the superscript is negative: sin²(x) is shorthand for sin(x).sin(x).

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.