In mathematics, the composition operator <math>\circ</math> takes two functions, <math>f</math> and <math>g</math>, and returns a new function <math>f \circ g</math>. When the composite function <math>f \circ g</math> (pronounced "<math>f</math> of <math>g</math>") is evaluated at an input <math>x</math>, the result is <math>(f \circ g)(x) = f(g(x))</math>. That is, the function <math>f</math> is applied after applying <math>g</math> to <math>x</math>.

The composition of functions is a special case of the composition of relations, sometimes also denoted by <math>\circ</math>. As a result, all properties of composition of relations are true of composition of functions, Since the parentheses do not change the result, they are generally omitted.

In a strict sense, the composition is only meaningful if the codomain of equals the domain of ; in a wider sense, it is sufficient that the former be an improper subset of the latter.

Composition monoids

Suppose one has two (or more) functions having the same domain and codomain; these are often called transformations. Then one can form chains of transformations composed together, such as . Such chains have the algebraic structure of a monoid, called a transformation monoid or (much more seldom) a composition monoid. In general, transformation monoids can have remarkably complicated structure. One particular notable example is the de Rham curve. The set of all functions is called the full transformation semigroup</blockquote>

Typography

The composition symbol is encoded as ; see the Degree symbol article for similar-appearing Unicode characters. In TeX, it is written <code>\circ</code>.

See also

  • Cobweb plot – a graphical technique for functional composition
  • Combinatory logic
  • Composition ring, a formal axiomatization of the composition operation
  • Flow (mathematics)
  • Function composition (computer science)
  • Function of random variable, distribution of a function of a random variable
  • Functional decomposition
  • Functional square root
  • Functional equation
  • Higher-order function
  • Infinite compositions of analytic functions
  • Iterated function
  • Lambda calculus

Notes

References

  • "Composition of Functions" by Bruce Atwood, the Wolfram Demonstrations Project, 2007.