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Di/ck in the a/ss
{{about||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:X^4 - 4^x.PNG|350px|thumb|Graph of the function {{nowrap|1=''f''(''x'') = ''x''<sup>4</sup> − 4<sup>''x''</sup>}} over the interval [−2,+3]. Also shown are its two real roots and global minimum over the same interval.]]
In mathematics, the '''graph''' of a [[function (mathematics)|function]] ''f'' is the collection of all [[ordered pair]]s {{nowrap|(''x'', ''f''(''x''))}}. If the function input ''x'' is a [[Scalar (mathematics)|scalar]], the graph is a [[two-dimensional graph]], and for a [[continuous function]] is a [[curve]]. If the function input ''x'' is an ordered pair {{nowrap|(''x''<sub>1</sub>, ''x''<sub>2</sub>)}} of real numbers, the graph is the collection of all [[ordered triple]]s {{nowrap|(''x''<sub>1</sub>, ''x''<sub>2</sub>, ''f''(''x''<sub>1</sub>, ''x''<sub>2</sub>))}}, and for a continuous function is a [[Surface (topology)|surface]].
Informally, if ''x'' is a [[real number]] and ''f'' is a [[real function]], ''graph'' may mean the graphical representation of this collection, in the form of a [[line chart]]: a [[curve]] on a [[Cartesian coordinate system|Cartesian plane]], together with Cartesian axes, etc. Graphing on a Cartesian plane is sometimes referred to as ''curve sketching''. The graph of a function on real numbers may be mapped directly to the graphic representation of the function. For general functions, a graphic representation cannot necessarily be found 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 generalized 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 that two functions with different [[codomain]] could have the same graph. For example, the cubic polynomial mentioned below is a [[surjection]] if its codomain is the [[real number]]s but it is not if its codomain is the [[complex number|complex field]].
To test whether a graph of a [[curve]] is a [[Function (mathematics)|function]] of ''x'', one uses the [[vertical line test]]. To test whether a graph of a curve is a function of ''y'', one uses the [[horizontal line test]]. If the function has an inverse, the graph of the inverse can be found by reflecting the graph of the original function over the line {{nowrap|1=''y'' = ''x''}}.
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.
In the modern [[foundation of mathematics]] known as [[set theory]], a function and its graph are essentially the same thing.<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|origyear=1971|publisher=Dover Publications|isbn=978-0-486-79549-2|pages=49}}</ref>
[[File:F(x) = x^3 − 9x.PNG|right|thumb|250px| Graph of the function {{nowrap|1=''f''(''x'') = ''x''<sup>3</sup> − 9''x''}}]]
== Examples ==
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