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set theory > function composition
number > elementary arithmetic > operation > binary operation > function composition
algebra > elementary algebra > operation > binary operation > function composition

Preferred term

function composition  

Definition

  • 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.
    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, 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.

    (Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Function_composition)

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