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'{{two other uses||the combinatorial structure|Graph (mathematics)|the graph-theoretic representation of a function from a set to itself|Functional graph}} {{merge|Plot (graphics)|target=Graph (plot)|discuss=Talk:Graph of a function#Requested merge and move 14 January 2016|date=January 2016}} {{refimprove|date=August 2014}} [[File:X^4 - 4^x.PNG|350px|thumb|Graph of the function {{nowrap|1=''f''(''x'') = ''x''⁴ − 4^''x''}} 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''₁, ''x''₂)}} of real numbers, the graph is the collection of all [[ordered triple]]s {{nowrap|(''x''₁, ''x''₂, ''f''(''x''₁, ''x''₂))}}, and for a continuous function is a [[surface]] (see [[three-dimensional graph]]). 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''³ − 9''x''}}]] == Examples == === Functions of one variable === [[File:Three-dimensional graph.png|right|thumb|250px|Graph of the [[function (mathematics)|function]] {{nowrap|1=''f''(''x'', ''y'') = sin(''x''²) · cos(''y''²)}}.]] 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''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}} === Functions of two variables === [[File:F(x,y)=−((cosx)^2 + (cosy)^2)^2.PNG|thumb|250px|Plot of the graph of {{nowrap|1=''f''(''x'', ''y'') = −(cos(''x''²) + cos(''y''²))²}}, also showing its gradient projected on the bottom plane.]] The graph of the [[trigonometric function]] : ''f''(''x'', ''y'') = sin(''x''²) · cos(''y''²) is : { (''x'', ''y'', sin(''x''²) · cos(''y''²)) : ''x'' and ''y'' are real numbers }. 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). Oftentimes it is helpful to show with the graph, the gradient of the function and several level curves. The level curves can be mapped on the function surface or can be projected on the bottom plane. The second figure shows such a drawing of the graph of the function: : ''f''(''x'', ''y'') = −(cos(''x''²) + cos(''y''²))² === Normal to a graph === Given a function ''f'' of ''n'' variables: <math> x=x_1, \dotsc, x_n </math>, the [[normal (geometry)|normal]] to the graph is : <math>(\nabla f, -1) </math> (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. == Generalizations == 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]]. == Tools for plotting function graphs == === Hardware === * [[Graphing calculator]] * [[Oscilloscope]] === Software === See [[List of graphing software]] == See also == {{colbegin||25em}} * [[Asymptote]] * [[Chart]] * [[Concave function]] * [[Convex function]] * [[Contour line|Contour plot]] * [[Critical point (mathematics)|Critical point]] * [[Derivative]] * [[Epigraph (mathematics)|Epigraph]] * [[Slope]] * [[Solution point]] * [[Stationary point]] * [[Tetraview]] * [[Vertical translation]] * [[Y-intercept]] {{colend}} ==References== {{reflist}} == External links == {{Commons category|Graphs}} * [http://pedritoclavito.netau.net/graphics2/graph.html Graph of function, derivative and antiderivative plotter] * Weisstein, Eric W. "[http://mathworld.wolfram.com/FunctionGraph.html Function Graph]." From MathWorld—A Wolfram Web Resource. {{Visualization}} [[Category:Charts]] [[Category:Functions and mappings]] [[pt:Função#Gráficos de função]]'