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*". *g*o*f*, 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* o*f*" 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` o`f`^{ 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: sin^{2}(*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 |