Editing Graph of a function (section)

{{Short description|Representation of a mathematical function}}
{{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}}

In [[mathematics]], the '''graph of a function''' <math>f</math> is the set of [[ordered pair]]s <math>(x, y)</math>, where <math>f(x) = y.</math> In the common case where <math>x</math> and <math>f(x)</math> are [[real number]]s, these pairs are [[Cartesian coordinates]] of points in a [[plane (geometry)|plane]] and often form a [[Plane curve|curve]].
The graphical representation of the graph of a [[Function (mathematics)|function]] is also known as a ''[[Plot (graphics)|plot]]''.

In the case of [[Bivariate function|functions of two variables]] – that is, functions whose [[Domain of a function|domain]] consists of pairs <math>(x, y)</math> –, the graph usually refers to the set of [[ordered triple]]s <math>(x, y, z)</math> where <math>f(x,y) = z</math>. This is a subset of [[three-dimensional space]]; for a continuous [[real-valued function]] of two real variables, its graph forms a [[Surface (mathematics)|surface]], which can be visualized as a ''[[surface plot (graphics)|surface plot]]''.

In [[science]], [[engineering]], [[technology]], [[finance]], and other areas, graphs are tools used for many purposes. In the simplest case one variable is plotted as a function of another, typically using [[Rectangular coordinate system|rectangular axes]]; see ''[[Plot (graphics)]]'' for details.

{{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.]]