Sigmoid function - Revision history

31.7.76.171 at 07:06, 17 November 2014

2014-11-17T07:06:12Z

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en>Sbmehta: capitalization

2014-03-02T23:58:35Z

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en>Monkbot: /* Examples */Fix CS1 deprecated date parameter errors

2014-01-30T01:42:26Z

Examples: Fix CS1 deprecated date parameter errors

← Older revision		Revision as of 03:42, 30 January 2014
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	~~My name~~ is ~~Ethan Luther~~. ~~I life~~ in ~~Lake Wales~~ (~~United States~~).<br><br>~~Here~~ is ~~my blog post~~ [http://~~safedietsthatwork~~.~~webs~~.com ~~diet plans~~]		{{for\|the graph-theoretic representation of a function from a set to the same set\|Functional graph}}
			[[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)]]
			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]]).

			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'', 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''}}.

			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 {{nowrap\|1=''f''(''x'') = ''x''<sup>3</sup> − 9''x''}}]]

			=== Functions of one variable ===
			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 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]] {{nowrap\|1=''f''(''x'', ''y'') = sin(''x''<sup>2</sup>) · cos(''y''<sup>2</sup>)}}.]]

			=== Functions of two variables ===

			The graph of the [[trigonometric]] function on the real line
			: ''f''(''x'', ''y'') = [[sine\|sin]](''x''<sup>2</sup>) · [[cosine\|cos]](''y''<sup>2</sup>)
			is
			: {(''x'', ''y'', sin(''x''<sup>2</sup>) · cos(''y''<sup>2</sup>)) : ''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).
			{{clear}}

			=== Functions of one variable ===
			You can see this set on a two dimensional cartesian coordinate system {{nowrap\|(''x'', ''y'')}}, using color to display the third coordinate ''z''.
			<gallery>
			Image:Sinx2+siny2.jmb.jpg\|{{nowrap\|1=''z'' = sin(''x'')<sup>2</sup> + sin(''y'')<sup>2</sup>}}
			</gallery>

			=== Normal to a graph ===
			Given a function ''f'' of ''n'' variables: <math> x=x_1, \dotsc, x_n </math>, the 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]]
			* [[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]]
			* [[Graph theory]]
			</div>

			== 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]]

en>Wdanwatts: Reference

2012-08-21T15:58:37Z

Reference

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My name is Ethan Luther. I life in Lake Wales (United States).<br><br>Here is my blog post [http://safedietsthatwork.webs.com diet plans]