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• Function_composition abstract "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 to yield a function which maps x in X to g(f(x)) in Z. Intuitively, if z is a function g of y and y is a function f of x, then z is a function of x.Composing functions is a chaining process using two functions in which the output of the first function becomes the input of the second function. The resulting composite function, notated g ∘ f : X → Z --- interchangeably written, in many sources and within this article, as g ∘ f : X → Y --- is 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 "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 left off without causing any ambiguity.When composing two functions, it is important to consider the domains of both functions. Therefore, the domain of the output reflects both functions. In most cases, the domain for a composition restricts any inputs that are not allowed in the domain of the first function. Additionally, the domain restricts any values whose output from the first function will be restricted by the second function. For example, consider the functions f(x)=sqrt(x) and g(x)=x². The domain of f(x) is all non-negative real numbers. This is because the result of a negative integer under the radical will result in an imaginary number. The domain of g(x) is all real numbers. This is because any real number can be squared and produces a continuous function. The resulting composition is: h(x) = (g o f)(x) = g(f(x)) = sqrt(x²) = x. Therefore, the composite function h(x) = x. Usually this function has the domain of all real numbers. But since the non-negative real numbers is a subset of the real numbers, and the non-negative real numbers is the domain of f(x), the domain of h(x) therefore has to be the subset, which is all non-negative real numbers. 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.The composition of one-to-one functions is always one-to-one. Given two functions f(a) and g(a), both of which are one-to-one, or injective, the composition of these functions (f o g)(a) = f(g(a)) then results in a one-to-one function. Proof: Our goal is to show that (g o f)(a) is one-to-one. Let X, Y, and Z be sets. Let f: X → Y and g: Y→ Z be functions. Assume both f and g are one-to-one. Choose a and b in X and assume (g o f)(a) = (g o f)(b). By the definition of composite functions, g(f(a)) = g(f(b)). Since g is one-to-one, f(a) = f(b). Thus, since f is also one-to-one, we know that a = b. Therefore, (g o f)(a) is one-to-one. The composition of two onto functions is always onto. Given two functions f(a) and g(a), both of which are onto, or surjective,the composition of these functions (f o g)(a) = f(g(a)) then results in an onto function. Proof: Our goal is show that (f o g)(a) = f(g(a)) is onto. Let f: X → Y and g: Y→ Z be functions. Assume both f and g are onto. Let Z be a set that includes the arbitrary element c. Since f is onto, there is an element b in Y such that c = f(b). Additionally, since g is onto, there is an element a in X such that b = g(a). Then, by the definition of composite functions, c = f(g(a)) = (f o g)(a).".
• Function_composition thumbnail Compfun.svg?width=300.
• Function_composition wikiPageExternalLink CompositionOfFunctions.
• Function_composition wikiPageID "195947".
• Function_composition wikiPageRevisionID "606578578".
• Function_composition hasPhotoCollection Function_composition.
• Function_composition id "p/c024260".
• Function_composition title "Composite function".
• Function_composition subject Category:Basic_concepts_in_set_theory.
• Function_composition subject Category:Binary_operations.
• Function_composition subject Category:Functions_and_mappings.
• Function_composition type Abstraction100002137.
• Function_composition type BasicConceptsInSetTheory.
• Function_composition type Cognition100023271.
• Function_composition type Concept105835747.
• Function_composition type Content105809192.
• Function_composition type Function113783816.
• Function_composition type FunctionsAndMappings.
• Function_composition type Idea105833840.
• Function_composition type MathematicalRelation113783581.
• Function_composition type PsychologicalFeature100023100.
• Function_composition type Relation100031921.
• Function_composition comment "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 to yield a function which maps x in X to g(f(x)) in Z. Intuitively, if z is a function g of y and y is a function f of x, then z is a function of x.Composing functions is a chaining process using two functions in which the output of the first function becomes the input of the second function.".
• Function_composition label "Composition de fonctions".
• Function_composition label "Composizione di funzioni".
• Function_composition label "Composição de funções".
• Function_composition label "Función compuesta".
• Function_composition label "Functiecompositie".
• Function_composition label "Function composition".
• Function_composition label "Komposition (Mathematik)".
• Function_composition label "Złożenie funkcji".
• Function_composition label "Композиция функций".
• Function_composition label "تركيب الدوال".
• Function_composition label "写像の合成".
• Function_composition label "复合函数".
• Function_composition sameAs Skládání_zobrazení.
• Function_composition sameAs Komposition_(Mathematik).
• Function_composition sameAs Σύνθεση_συνάρτησης.
• Function_composition sameAs Función_compuesta.
• Function_composition sameAs Composition_de_fonctions.
• Function_composition sameAs Composizione_di_funzioni.
• Function_composition sameAs 写像の合成.
• Function_composition sameAs 합성함수.
• Function_composition sameAs Functiecompositie.
• Function_composition sameAs Złożenie_funkcji.
• Function_composition sameAs Composição_de_funções.
• Function_composition sameAs m.01bvfb.
• Function_composition sameAs Q244761.
• Function_composition sameAs Q244761.
• Function_composition sameAs Function_composition.
• Function_composition wasDerivedFrom Function_composition?oldid=606578578.
• Function_composition depiction Compfun.svg.
• Function_composition isPrimaryTopicOf Function_composition.