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Reverted 1 edit by Farkle Griffen (talk): In most sources, the graph is the set of the pairs, and it is distinguished from its plot (see the end of the paragraph) |
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{{About||graph-theoretic representation of a function|Functional graph}}
{{more citations needed|date=August 2014}}
[[File:Polynomial of degree three.svg|class=skin-invert-image|thumb|250x250px|Graph of the function <math>f(x)=\frac{x^3+3x^2-6x-8}{4}.</math>]]
{{functions}}
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{{anchor|graph of a relation}}A graph of a function is a special case of a [[Relation (mathematics)|relation]].
In the modern [[foundations of mathematics]], and, typically, in [[set theory]], a function is actually equal to its graph.<ref name="Pinter2014">{{cite book|author=Charles C Pinter|title=A Book of Set Theory|url=https://books.google.com/books?id=iUT_AwAAQBAJ&pg=PA49|year=2014|orig-year=1971|publisher=Dover Publications|isbn=978-0-486-79549-2|pages=49}}</ref> However, it is often useful to see functions as [[Map (mathematics)|mappings]],<ref>{{cite book|author=T. M. Apostol|authorlink=Tom M. Apostol|title=Mathematical Analysis|year=1981|publisher=Addison-Wesley|page=35}}</ref> which consist not only of the relation between input and output, but also which set is the ___domain, and which set is the [[codomain]]. For example, to say that a function is onto ([[Surjective function|surjective]]) or not the codomain should be taken into account. The graph of a function on its own does not determine the codomain. It is common<ref>{{cite book|author=P. R. Halmos|title=A Hilbert Space Problem Book|url=https://archive.org/details/hilbertspaceprob00halm_811|url-access=limited|year=1982|publisher=Springer-Verlag|isbn=0-387-90685-1|page=[https://archive.org/details/hilbertspaceprob00halm_811/page/n47 31]}}</ref> to use both terms ''function'' and ''graph of a function'' since even if considered the same object, they indicate viewing it from a different perspective.
[[File:X^4 - 4^x.PNG|class=skin-invert-image|350px|thumb|Graph of the function <math>f(x) = x^4 - 4^x</math> over the [[Interval (mathematics)|interval]] [−2,+3]. Also shown are the two real roots and the local minimum that are in the interval.]]
== Definition ==
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Given a [[function (mathematics)|function]] <math>f : X \to Y</math> from a set {{mvar|X}} (the [[___domain of a function|___domain]]) to a set {{mvar|Y}} (the [[codomain]]), the graph of the function is the set<ref>{{cite book|author=D. S. Bridges|title=Foundations of Real and Abstract Analysis|url=https://archive.org/details/springer_10.1007-978-0-387-22620-0|year=1991|publisher=Springer|page=[https://archive.org/details/springer_10.1007-978-0-387-22620-0/page/n292 285]|isbn=0-387-98239-6}}</ref>
<math display=block>G(f) = \{(x,f(x)) : x \in X\},</math>
which is a subset of the [[Cartesian product]] <math>X\times Y</math>. In the
== Examples ==
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=== Functions of two variables ===
[[File:F(x,y)=−((cosx)^2 + (cosy)^2)^2.PNG|class=skin-invert-image|thumb|250px|Plot of the graph of <math>f(x, y) = - \left(\cos\left(x^2\right) + \cos\left(y^2\right)\right)^2,</math> also showing its gradient projected on the bottom plane.]]
The graph of the [[trigonometric function]]
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