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'{{for|the graph-theoretic representation of a function from a set to the same set|Functional graph}} In mathematics, the '''graph''' of a [[function (mathematics)|function]] ''f'' is the collection of all [[ordered pair]]s (''x'', ''f''(''x'')). In particular, if ''x'' is a [[real number]], ''graph'' means the graphical representation of this collection, in the form of 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'''. If the function input ''x'' is an ordered pair (''x''₁, ''x''₂) of real numbers, the graph is the collection of all [[ordered triple]]s (''x''₁, ''x''₂, ''f''(''x''₁, ''x''₂)), and its graphical representation is a [[surface]] (see [[three dimensional graph]]). 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 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 if a graph of a [[curve]] is a [[Function (mathematics)|function]], use the [[vertical line test]]. To test if the function is [[one-to-one function|one-to-one]], meaning it has an [[inverse|inverse function]], use 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 <math>y = x</math>. A curve is a one-to-one function [[if and only if]] it is a function and it passes the horizontal line test. 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. == Examples == [[Image:cubicpoly.png||right|thumb|300 px| Graph of the function ''f''(''x'')=''x''³ - 9''x'']] === Functions of one variable === The graph of the function. : <math>f(x)= \left\{\begin{matrix} 0, & \mbox{if }x<0 \\ 2x, & \mbox{if }0<x<1 \\ 0, & \mbox{if }x>1. \end{matrix}\right. </math> is :{(x<0,0), (0<x<0,2x), (x>1,0)}. The graph of the cubic polynomial on the [[one variable]] : <math>f(x)={(x<0,0)} \!\ </math> is : {(''x'', ''x''³-9''x'') : ''x'' is a real number}. If this set is plotted on a Cartesian plane, the result is a curve (see figure). {{clear}} [[image:Three-dimensional graph.png|right|thumb|300px|Graph of the [[function (mathematics)|function]] ''f(x, y) = [[sine|sin]](x²)·[[cosine|cos]](y²)''.]] === Functions of two variables === The graph of the [[trigonometric]] function on the real line : ''f(x, y) = [[sine|sin]](x²)·[[cosine|cos]](y²)'' is : {(''x'', ''y'', [[sine|sin]](x²)·[[cosine|cos]](y²)) : ''x'' is a real number}. If this set is plotted on a three dimensional Cartesian coordinate system, the result is a surface (see figure). {{clear}} == Tools for plotting function graphs == === [[Hardware]] === * [[Graphing calculator]] * [[Oscilloscope]] * [[Paper]] and [[pencil]] === [[Software]] === See [[List of graphing software]] == See also == <div style="-moz-column-count:2; column-count:2;"> * [[Asymptote]] * [[Critical point (mathematics)|Critical point]] * [[Derivative]] * [[Epigraph (mathematics)|Epigraph]] * [[Chart]] * [[Stationary point]] * [[Slope]] * [[Solution point]] * [[Tetraview]] * [[Vertical translation]] * [[Y-intercept]] </div> == External links == {{Commonscat|Graphs}} * Weisstein, Eric W. "[http://mathworld.wolfram.com/FunctionGraph.html Function Graph]." From MathWorld--A Wolfram Web Resource. {{Visualization}} [[Category:Charts]] [[Category:Functions and mappings]] [[ar:رسم بياني لدالة]] [[bs:Grafik funkcije]] [[ca:Gràfica d'una funció]] [[cs:Graf (funkce)]] [[de:Funktionsgraph]] [[el:Γραφική παράσταση συνάρτησης]] [[es:Gráfica de una función]] [[eo:Grafikaĵo]] [[eu:Funtzio baten irudikapen grafiko]] [[fr:Graphe d'une fonction]] [[ko:함수의 그래프]] [[is:Línurit]] [[it:Grafico di una funzione]] [[he:גרף של פונקציה]] [[lo:ເສັ້ນສະແດງ]] [[la:Graphum (mathematica)]] [[hu:Grafikon (matematika)]] [[nl:Grafiek (wiskunde)]] [[ja:グラフ (関数)]] [[no:Funksjonsgrafen]] [[pl:Wykres funkcji]] [[pt:Função#Gráficos de função]] [[ru:График функции]] [[simple:Graph]] [[sk:Graf funkcie]] [[sl:Graf funkcije]] [[sv:Linjediagram]] [[th:กราฟของฟังก์ชัน]] [[uk:Графік функції]] [[ur:Graph of a function]] [[zh:函数图像]]'