Function |
---|
x ↦ f (x) |
History of the function concept |
Examples of domains and codomains |
Classes/properties |
Constructions |
Generalizations |
In mathematics, function composition is an operation ∘ that takes two functions f and g, and produces a function h = g ∘ f such that h(x) = g(f(x)). In this operation, the function g is applied to the result of applying the function f to x. That is, the functions f : X → Y and g : Y → Z are composed to yield a function that maps x in domain X to g(f(x)) in codomain Z. Intuitively, if z is a function of y, and y is a function of x, then z is a function of x. The resulting composite function is denoted g ∘ f : X → Z, defined by (g ∘ f )(x) = g(f(x)) for all x in X.[nb 1]
The notation g ∘ f is read as "g of f ", "g after f ", "g circle f ", "g round f ", "g about f ", "g composed with f ", "g following f ", "f then g", or "g on f ", or "the composition of g and f ". Intuitively, composing functions is a chaining process in which the output of function f feeds the input of function g.
The composition of functions is a special case of the composition of relations, sometimes also denoted by . As a result, all properties of composition of relations are true of composition of functions,[1] such as the property of associativity.
Composition of functions is different from multiplication of functions (if defined at all), and has some quite different properties; in particular, composition of functions is not commutative.[2]
Cite error: There are <ref group=nb>
tags on this page, but the references will not show without a {{reflist|group=nb}}
template (see the help page).
Velleman_2006
was invoked but never defined (see the help page).© MMXXIII Rich X Search. We shall prevail. All rights reserved. Rich X Search