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Concept# Function composition

Summary

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.
Composition of functions on a finite set: If f = {(1, 1), (2, 3), (3, 1), (4, 2)} , and g = {(1, 2), (2, 3), (3, 1), (4, 2)} , then g ∘ f = {(1, 2), (2, 1), (3, 2), (4, 3)} , as shown in the figure.
Composition of functions on an infinite set: If f: R → R (where R is the set of all real numbers) is given by f(x) = 2x + 4 and g: R → R is given by g(x) = x3, then:
If an airplane's altitude at time t is a(t), and the air pressure at altitude x is p(x), then (p ∘ a)(t) is the pressure around the plane at time t.
The composition of functions is always associative—a property inherited from the composition of relations. That is, if f, g, and h are composable, then f ∘ (g ∘ h) = (f ∘ g) ∘ h.

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