Function composition
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For function composition in computer science, see function composition (computer science).
In mathematics, a composite function, formed by the composition of one function on another, represents the application of the former to the result of the application of the latter to the argument of the composite. The functions f: X → Y and g: Y → Z can be composed by first applying f to an argument x and then applying g to the result. Thus one obtains a function g o f: X → Z defined by (g o f)(x) = g(f(x)) for all x in X. The notation g o f is read as "g circle f", or "g composed with 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 o (g o h) = (f o g) o h. Since there is no distinction between the choices of placement of parentheses, they may be safely left off. The functions g and f commute with each other if g o f = f o 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, Failed to parse (Missing texvc executable; please see math/README to configure.): \left | x \right | + 3 = \left | x + 3 \right |\, only when Failed to parse (Missing texvc executable; please see math/README to configure.): x \ge 0 . But inverse functions always commute to produce the identity mapping. 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.
ExampleAs 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 o h)(t) describes the oxygen concentration around the plane at time t. Functional powersIf Failed to parse (Missing texvc executable; please see math/README to configure.): Y \subseteq X then Failed to parse (Missing texvc executable; please see math/README to configure.): f: X\rightarrow Y may compose with itself; this is sometimes denoted Failed to parse (Missing texvc executable; please see math/README to configure.): f^2\, . Thus:
The functional powers Failed to parse (Missing texvc executable; please see math/README to configure.): f\circ f^n=f^n\circ f=f^{n+1} for natural Failed to parse (Missing texvc executable; please see math/README to configure.): n\, follow immediately.
Failed to parse (Missing texvc executable; please see math/README to configure.): \big( the identity map on the domain of Failed to parse (Missing texvc executable; please see math/README to configure.): f\big) .
admits an inverse function, negative functional powers Failed to parse (Missing texvc executable; please see math/README to configure.): f^{-k}\, Failed to parse (Missing texvc executable; please see math/README to configure.): (k>0\,) are defined as the opposite power of the inverse function, Failed to parse (Missing texvc executable; please see math/README to configure.): (f^{-1})^k\, . 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. A simple example would be that where f is the successor function, f r(x) = x + r. Iterated functions occur naturally in the study of fractals and dynamical systems. Composition monoidsSuppose one has two (or more) functions f: X → X, g: X → X having the same domain and range. Then one can form long, potentially complicated chains of these functions composed together, such as f o f o g o f. Such long chains have the algebraic structure of a monoid, sometimes called the 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 form a group; and one says that the group is generated by these functions. The set of all bijective functions f: X → X form a group with respect to the composition operator. This is the symmetric group, also sometimes called the composition group. Alternative notationIn the mid-20th century, some mathematicians decided that writing "g o f" to mean "first apply f, then apply g" was too confusing and decided to change notations. They wrote "xf" for "f(x)" and "xfg" for "g(f(x))". This can be more natural and seem simpler than writing functions on the left in some areas. Category Theory uses f;g interchangeably with g o f. Composition operatorGiven a function g, the composition operator Failed to parse (Missing texvc executable; please see math/README to configure.): C_g is defined as that operator which maps functions to functions as
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External links
ca:Composició funcional cs:Skládání zobrazení da:Sammensat funktion de:Komposition (Mathematik) fr:Composition de fonctions hr:Kompozicija funkcija it:Composizione di funzioni he:הרכבת פונקציות pl:Złożenie funkcji pt:Composição de funções ru:Композиция функций sl:Kompozitum funkcij sv:sammansatt funktion fi:Yhdistetty funktio uk:Композиція функцій |
