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| {{for|the graph-theoretic representation of a function from a set to the same set|Functional graph}}
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| [[File:X^4-4^x.gif|2900px|thumbnail|right|Graph of the function {{nowrap|1=''f''(''x'') = ''x''<sup>4</sup> − 4<sup>''x''</sup>}} {{-}}(−2, +2)]]
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| 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 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]] (see [[three-dimensional graph]]).
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| 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]].
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| 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]].
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| To test whether a graph of a [[curve]] is a [[Function (mathematics)|function]] of ''x'', use the [[vertical line test]]. To test whether a graph of a curve is a function of ''y'', 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 {{nowrap|1=''y'' = ''x''}}.
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| 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.
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| == Examples ==
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| [[Image:cubicpoly.png||right|thumb|300 px| Graph of the function {{nowrap|1=''f''(''x'') = ''x''<sup>3</sup> − 9''x''}}]]
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| === Functions of one variable ===
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| The graph of the function.
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| : <math>f(x)=
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| \left\{\begin{matrix}
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| a, & \mbox{if }x=1 \\ d, & \mbox{if }x=2 \\ c, & \mbox{if }x=3.
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| \end{matrix}\right.
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| </math>
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| is
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| :{(1,a), (2,d), (3,c)}.
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| The graph of the cubic polynomial on the [[real line]]
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| : <math>f(x) = x^3 - 9x</math>
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| is
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| : {(''x'', ''x''<sup>3</sup> − 9''x'') : ''x'' is a real number}.
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| If this set is plotted on a Cartesian plane, the result is a curve (see figure).
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| {{clear}}
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| [[image:Three-dimensional graph.png|right|thumb|300px|Graph of the [[function (mathematics)|function]] {{nowrap|1=''f''(''x'', ''y'') = sin(''x''<sup>2</sup>) · cos(''y''<sup>2</sup>)}}.]]
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| === Functions of two variables ===
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| The graph of the [[trigonometric]] function on the real line
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| : ''f''(''x'', ''y'') = [[sine|sin]](''x''<sup>2</sup>) · [[cosine|cos]](''y''<sup>2</sup>)
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| is
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| : {(''x'', ''y'', sin(''x''<sup>2</sup>) · cos(''y''<sup>2</sup>)) : ''x'' and ''y'' are real numbers}.
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| If this set is plotted on a [[Cartesian_coordinate_system#Cartesian_coordinates_in_three_dimensions|three dimensional Cartesian coordinate system]], the result is a surface (see figure).
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| {{clear}}
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| === Functions of one variable ===
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| You can see this set on a two dimensional cartesian coordinate system {{nowrap|(''x'', ''y'')}}, using color to display the third coordinate ''z''.
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| <gallery>
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| Image:Sinx2+siny2.jmb.jpg|{{nowrap|1=''z'' = sin(''x'')<sup>2</sup> + sin(''y'')<sup>2</sup>}}
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| </gallery>
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| === Normal to a graph ===
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| Given a function ''f'' of ''n'' variables: <math> x=x_1, \dotsc, x_n </math>, the normal to the graph is
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| : <math>(\nabla f, -1) </math>
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| (up to multiplication by a constant). This is seen by considering the graph as a [[level set]] of the function <math>g(x,z) = f(x) - z</math>, and using that <math>\nabla g </math> is normal to the level sets.
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| == Generalizations ==
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| The graph of a function is contained in a [[cartesian product]] of sets. An X–Y plane is a cartesian product of two lines, called X and Y, while a cylinder is a cartesian product of a line and a circle, whose height, radius, and angle assign precise locations of the points. [[Fibre bundle]]s aren't cartesian products, but appear to be up close. There is a corresponding notion of a graph on a fibre bundle called a [[Section (fiber bundle)|section]].
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| == Tools for plotting function graphs ==
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| === Hardware ===
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| * [[Graphing calculator]]
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| * [[Oscilloscope]]
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| * [[Paper]] and [[pencil]]
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| === Software ===
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| See [[List of graphing software]]
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| == See also ==
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| <div style="-moz-column-count:2; column-count:2;">
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| * [[Asymptote]]
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| * [[Critical point (mathematics)|Critical point]]
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| * [[Derivative]]
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| * [[Epigraph (mathematics)|Epigraph]]
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| * [[Chart]]
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| * [[Stationary point]]
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| * [[Slope]]
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| * [[Solution point]]
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| * [[Tetraview]]
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| * [[Vertical translation]]
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| * [[Y-intercept]]
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| * [[Graph theory]]
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| </div>
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| == External links ==
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| {{Commons category|Graphs}}
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| * [http://pedritoclavito.netau.net/graphics2/graph.html Graph of function, derivative and antiderivative plotter]
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| * Weisstein, Eric W. "[http://mathworld.wolfram.com/FunctionGraph.html Function Graph]." From MathWorld—A Wolfram Web Resource.
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| {{Visualization}}
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| [[Category:Charts]]
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| [[Category:Functions and mappings]]
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| [[pt:Função#Gráficos de função]]
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Ed is what individuals contact me and my spouse doesn't like it at all. To play lacross is the thing I love most of all. My working day occupation is an invoicing officer but I've already utilized for an additional one. My wife and I reside in Mississippi but now I'm contemplating other options.
My homepage :: phone psychic; http://www.monsterhuntclips.com/,