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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 enlarged the domains of application of the concept.
A function is most often denoted by letters such as {{mvar|f}}, {{mvar|g}} and {{mvar|h}}, and the value of a function {{mvar|f}} at an element {{mvar|x}} of its ___domain is denoted by {{math|''f''(''x'')}}; the numerical value resulting from the '''{{vanchor|function evaluation}}''' at a particular input value is denoted by replacing {{mvar|x}} with this value; for example, the value of {{mvar|f}} at {{math|''x'' {{=}} 4}} is denoted by {{math|''f''(4)}}. When the function is not named and is represented by an [[expression (mathematics)|expression]] {{mvar|E}}, the value of the function at, say, {{math|''x'' {{=}} 4}} may be denoted by {{math|1={{itco|''E''}}{{!}}<sub>''x''=4</sub>}}. For example, the value at {{math|4}} of the function that maps {{mvar|x}} to <math>(x+1)^2</math> may be denoted by <math>\left.(x+1)^2\right\vert_{x=4}
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 web|title=function {{!}} Definition, Types, Examples, & Facts| url=https://www.britannica.com/science/function-mathematics|access-date=2020-08-17|website=Encyclopedia 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.
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