In mathematics, function composition is the pointwise application of one function to another to produce a third function. For instance, the functions f : X → Y and g : Y → Z can be composed by computing the output of g when it has an argument of f, and regarding this as a function of x. Intuitively, if z is a function g of y and y is a function f of x, then z is a function of x. Thus one obtains a composite function 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 circle f", or "g round f", or "g composed with f", "g after f", "g following f", or just "g of f". The composition of functions is always associative. That is, if f, g, and h are three functions with suitably chosen domains and codomains, then f ∘ (g ∘ h) = (f ∘ g) ∘ h, where the parentheses serve to indicate that composition is to be performed first for the parenthesized functions. Since there is no distinction between the choices of placement of parentheses, they may be safely left off. The functions g and f are said to commute with each other if g ∘ f = f ∘ g. In general, composition of functions will not be commutative. Commutativity is a special property, attained only by particular functions, and often in special circumstances. For example, |x| + 3 = |x + 3| only when x ≥ 0. Derivatives of compositions involving differentiable functions can be found using the chain rule. Higher derivatives of such functions are given by Faà di Bruno's formula. Example As an example, suppose that an airplane's elevation at time t is given by the function h(t) and that the oxygen concentration at elevation x is given by the function c(x). Then (c ∘ h)(t) describes the oxygen concentration around the plane at time t. Functional powers If then may compose with itself; this is sometimes denoted as f 2. Thus: Repeated composition of a function with itself is called iterated function. The functional powers for natural n follow immediately. By convention, the identity map on the domain of f ). If admits an inverse function, negative functional powers are defined as the opposite power of the inverse function,. Note: If f takes its values in a ring (in particular for real or complex-valued f), there is a risk of confusion, as f n could also stand for the n-fold product of f, e.g. f 2(x) = f(x) · f(x). For trigonometric functions, usually the latter is meant, at least for positive exponents. For example, in trigonometry, this superscript notation represents standard exponentiation when used with trigonometric functions: sin2(x) = sin(x) · sin(x). However, for negative exponents (especially −1), it nevertheless usually refers to the inverse function, e.g., tan−1 = arctan (≠ 1/tan). In some cases, an expression for f in g(x) = f r(x) can be derived from the rule for g given non-integer values of r. This is called fractional iteration. For instance, a half iterate of a function f is a function g satisfying g(g(x)) = f(x). Another example would be that where f is the successor function, f r(x) = x + r. This idea can be generalized so that the iteration count becomes a continuous parameter; in this case, such a system is called a flow, specified through solutions of Schröder's equation. Iterated functions and flows occur naturally in the study of fractals and dynamical systems. Composition monoids Suppose one has two (or more) functions f: X → X, g: X → X having the same domain and codomain. Then one can form long, potentially complicated chains of these functions composed together, such as f ∘ f ∘ g ∘ f. Such long chains have the algebraic structure of a monoid, called transformation monoid or composition monoid. In general, composition monoids can have remarkably complicated structure. One particular notable example is the de Rham curve. The set of all functions f: X → X is called the full transformation semigroup on X. If the functions are bijective, then the set of all possible combinations of these functions forms a transformation group; and one says that the group is generated by these functions. The set of all bijective functions f: X → X forms a group with respect to the composition operator. This is the symmetric group, also sometimes called the composition group. Alternative notations Many mathematicians omit the composition symbol, writing gf for g ∘ f. Attribution: Article text available under CC-BY-SA sourced from: http://en.wikipedia.org/wiki/Function_compositionPublic domain image sourced from http://wikipedia.org/wiki/File:Compfun.svg

• published: 18 Sep 2013
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