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In mathematics, the '''graph''' of a [[function (mathematics)|function]] is the collection of all pairs (''x'', ''f''(''x'')) of the function. In particular ''graph'' means the graphical representation of this collection, in the form of a curve, together with axes, etc. Graphing is sometimes referred to as '''curve sketching'''. 

The graph of the function
:<math>f(x)=\left\{\begin{matrix} a, & \mbox{if }x=1 \\ d, & \mbox{if }x=2 \\ c, & \mbox{if }x=3. \end{matrix}\right.</math>
is {(1,a), (2,d), (3,c)}.

The graph of the cubic polynomial on the real line
:<math>f(x)=x^3-9x</math>
is {(''x'',''x''<sup>3</sup>-9''x'') : ''x'' is a real number}.  If the set is plotted on a [[Cartesian coordinate system|Cartesian plane]], the result is

[[Image:cubicpoly.png]]  

Therefore the graph of a function on real numbers is identical to the graphic representation of the function. For general functions, the graphic representation cannot be applied and the formal definition of the graph of a function suits the need of mathematical statements, e.g., the [[closed graph theorem]] in [[functional analysis]].

The concept of the graph of a function is generalised to the graph of a [[relation (mathematics)|relation]].  Note that although a function is always identified with its graph, they are not the same because it will happen two functions with different [[codomain]] could have the same graph.  For example, the cubic polynomial mentioned above is a [[surjection]] if its codomain is the [[real number]]s but it is not if its codomain is the [[complex number|complex field]].

==Related articles ==
* [[concavity]]
* [[critical point]]
* [[derivative]]
* [[stationary point]]
* [[slope]]
* [[solution point]]
* [[vertical translation]]
* [[y-intercept]]

[[de:Funktionsgraph]]
[[eo:Grafeo]]
[[fr:Graphe]]
[[ja:&#12464;&#12521;&#12501;]]

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====References====
* Adapted from the [[Wikipedia]] article, "Graph_of_a_function" http://en.wikipedia.org/wiki/Graph_of_a_function, used under the [[GNU Free Documentation License]]