# 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

*g*is the parameter of

*f*, and the function

*f*is composed on

*g*. An equivalent representation is

*f*o

*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*o*f*, is customarily written *f* ^{2}. (*f*o*f*)(*x*)=*f*(*f*(*x*))=*f* ^{2}(*x*). Likewise, (*f*o*f*o*f*)(*x*)=*f*(*f*(*f*(*x*)))=*f* ^{3}(*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: sin^{2}(*x*) is shorthand for sin(*x*)*sin(*x*) or (sin(*x*) multiplied by 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.

See also: