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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)]]
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]]

Revision as of 00:58, 3 March 2014

Almost everyone is now aware which life-style illnesses are a leading killer in today's globe plus now the question on many women's mind is: are there diets which work for ladies? If you find yourself asking this question then you're not alone inside the quest for diets which work rapidly for women.

So, again, back to the all-natural aspect; most people have found diet plans effectively is changing your general food with organic replacements. These are clearly marked; frequently not containing pesticides, artificial flavoring, or preservatives. Doctors usually tell you that organic is the technique to go, however, several people create the mistake of thinking they will merely gorge themselves on organic food and everything is fine. That's no way to follow, even diets which work.

Similar to the 1200 calorie diet, a 1500 calorie diet plan is a diet that limits the amount of calories to be consumed inside a day, to 1500. Two such sample diet plans are given below inside a tabular shape.

Beachbody coaches furthermore sell a Shakeology shake with 280 calories, 34 grams of protein, 2 grams of fat and 6 grams of fiber. Shakeology has supports healthy blood glucose degrees similar to the Almased Diet Plan.

Calorie Restriction - how lose fat fast diets rely on limiting calories plus they fail to address insulin imbalance plus blood sugar fluctuations. What this means is by following the calorie limitation recommendations, you are hungry nearly all of the time. That's no method to lose weight forever. And the weight you do lose in the initially limited weeks is water plus lean muscle which sets the stage for more fat gain later. What you certainly like to do is to get rid of fat (not water plus muscle), you desire the reduction to be permanent and we do not need to be wanting food plus snacks frequently. You can read how to achieve this kind of fat loss at the moment.

The Cabbage Diet. Lose weight fast with all the cabbage diet. This diet consists of making a soup with cabbage, that you'll eat during the 7-day duration of the diet. The diet is low inside fat, plus high in fiber which can assist we lose weight swiftly. During this diet you'll rotate between eating cabbage soup plus fruit with eating greens and meat on every alternative day. This diet may assist you lose fat because it happens to be low inside calories. Some people find it hard to maintain this diet due to the limited food options it allows.

Learning that diets which function fast for women is a good first step in lucrative weight loss. Before you embark upon any diet plan it's best to recognize whether we need to lose fat. BMI for women will help we place into perspective your average fat thereby allowing we to better know whether a weight reduction system is right for we.