Graph of a function: Difference between revisions

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{{about||graphical representation|Plot (graphics)|the combinatorial structure|Graph (discrete mathematics)|the graph-theoretic representation of a function from a set to itself|Functional graph}}

{{refimprove|date=August 2014}}

[[File:F(x) = x^3 − 9x.PNG|right|thumb|250px| Graph of the function {{nowrap|1=''f''(''x'') = ''x''³ − 9''x''}}]]

In [[mathematics]], the '''graph''' of a [[function (mathematics)|function]] {{math|''f''}} is, formally, the set of all [[ordered pair]]s {{math|(''x'', ''f''(''x''))}}, such that {{math|''x''}} is in the [[Domain of a function|___domain of the function]] {{math|''f''}}. In the common case where {{mvar|x}} and {{math| ''f''(''x'')}} are [[real number|real numbers]], these pairs are [[Cartesian coordinates]] of points in the [[Euclidean plane]] and thus form a subset of this plane, which is a [[curve]] in the case of a [[continuous function]]. This graphical representation of the function is also called the ''graph of the function''.{{about|A page is a page. What more can I say?|graphical representation|Plot (graphics)|the combinatorial structure|Graph (discrete mathematics)|the graph-theoretic representation of a function from a set to itself|Functional graph}}In the case of functions of two variables, that is functions whose ___domain consists of pairs {{math|(''x'', ''y'')}}, the graph can be identified with the set of all [[ordered triple]]s {{math|(''x'', ''y'', ''f''(''x'', ''y''))}}. For a continuous [[real-valued function]] of two real variables, the graph is a [[Surface (mathematics)|surface]].

In the case of functions of two variables, that is functions whose ___domain consists of pairs {{math|(''x'', ''y'')}}, the graph can be identified with the set of all [[ordered triple]]s {{math|(''x'', ''y'', ''f''(''x'', ''y''))}}. For a continuous [[real-valued function]] of two real variables, the graph is a [[Surface (mathematics)|surface]].

{{anchor|graph of a relation}}

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== Definition ==

x square + y square = Two x y. Am I right?

Given a mapping <math>f:X \to Y</math>, in other words a function <math>f</math> together with its ___domain <math>X</math> and codomain <math>Y</math>, the graph of the mapping is<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=285|isbn=0-387-98239-6}}</ref> the set

:<math>G(f)=\{(x,f(x)) \mid x \in X\}</math>,

which is a subset of <math>X\times Y</math>. In the abstract definition of a function, <math>G(f)</math> is actually equal to <math>f</math>.

== Examples ==