quality:corruption, structural:scholar_edition
Function composition

Suppose the functions

or more compactly ξ = ξ (x), are all defined on a domain X. As the n -tuple x = (x, x, …, x) varies in X, a subset of R, the m -tuple ξ = (ξ, ξ, …, ξ) varies in another region Ξ a subset of R. To restate this:

Then, a function ζ of the functions ξ (x) defined on Ξ,

is a function composition defined on X, in other terms the mapping

Note the numbers m and n do not need to be equal.

For example, the function

defined everywhere on R can be rewritten by introducing

which is also everywhere defined in R to obtain

Function composition can be used to simplify functions, which is useful for carrying out multiple integrals and solving partial differential equations.