Editing Function (mathematics) (section)

{{Short description|Association of one output to each input}} 
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{{Functions}}

In [[mathematics]], a '''function''' from a [[set (mathematics)|set]] {{mvar|X}} to a set {{mvar|Y}} assigns to each element of {{mvar|X}} exactly one element of {{mvar|Y}}.<ref name=halmos>{{harvnb |Halmos |1970 |p=30}}; the words ''map'', ''mapping'', ''transformation'', ''correspondence'', and ''operator'' are sometimes used synonymously.</ref> The set {{mvar|X}} is called the [[Domain of a function|domain]] of the function<ref>{{harvnb|Halmos|1970}}</ref> and the set {{mvar|Y}} is called the [[codomain]] of the function.<ref name=codomain>{{eom|title=Mapping|oldid=37940|mode=cs1}}</ref>

Functions were originally the idealization of how a varying quantity depends on another quantity. For example, the position of a [[planet]] is a ''function'' of time. [[History of the function concept|Historically]], the concept was elaborated with the [[infinitesimal calculus]] at the end of the 17th century, and, until the 19th century, the functions that were considered were [[differentiable function|differentiable]] (that is, they had a high degree of regularity). The concept of a function was formalized at the end of the 19th century in terms of [[set theory]], and this greatly increased the possible applications of the concept.

A function is often denoted by a letter such as {{mvar|f}}, {{mvar|g}} or {{mvar|h}}. The value of a function {{mvar|f}} at an element {{mvar|x}} of its domain (that is, the element of the codomain that is associated with {{mvar|x}}) is denoted by {{math|''f''(''x'')}}; for example, the value of {{mvar|f}} at {{math|''x'' {{=}} 4}} is denoted by {{math|''f''(4)}}. Commonly, a specific function is defined by means of an [[expression (mathematics)|expression]] depending on {{mvar|x}}, such as <math>f(x)=x^2+1;</math> in this case, some computation, called '''{{vanchor|function evaluation}}''', may be needed for deducing the value of the function at a particular value; for example, if <math>f(x)=x^2+1,</math> then <math>f(4)=4^2+1=17.</math>

Given its domain and its codomain, a function is uniquely represented by the set of all [[pair (mathematics)|pairs]] {{math|(''x'', ''f''{{hair space}}(''x''))}}, called the ''[[graph of a function|graph of the function]]'', a popular means of illustrating the function.<ref group="note">This definition of "graph" refers to a ''set'' of pairs of objects. Graphs, in the sense of ''diagrams'', are most applicable to functions from the real numbers to themselves. All functions can be described by sets of pairs but it may not be practical to construct a diagram for functions between other sets (such as sets of matrices).</ref><ref>{{Cite encyclopedia|title=function {{!}} Definition, Types, Examples, & Facts| url=https://www.britannica.com/science/function-mathematics|access-date=2020-08-17|encyclopedia=Encyclopædia Britannica|language=en}}</ref> When the domain and the codomain are sets of real numbers, each such pair may be thought of as the [[Cartesian coordinates]] of a point in the plane.

Functions are widely used in [[science]], [[engineering]], and in most fields of mathematics. It has been said that functions are "the central objects of investigation" in most fields of mathematics.{{sfn |Spivak |2008 |p=39}}

The concept of a function has evolved significantly over centuries, from its informal origins in ancient mathematics to its formalization in the 19th century. See [[History of the function concept]] for details.