Graph of a function: Difference between revisions

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{{shortShort description|Representation of a mathematical function as the set of pairs (x, f(x))}}

{{aboutAbout||graphical representation|Plot (graphics)|the combinatorial structure|Graph (discrete mathematics)|the graph-theoretic representation of a function from a set to itself|Functional graph}}

{{refimprovemore citations needed|date=August 2014}}

[[File:F(x)Polynomial = x^3of −degree 9xthree.PNGsvg|rightclass=skin-invert-image|thumb|250px250x250px| Graph of the function {{nowrap|1=''<math>f''(''x'') = ''\frac{x''<sup>^3+3x^2-6x-8}{4}.</supmath> − 9''x''}}]]

{{functions}}

In [[mathematics]], the '''graph''' of a [[function''' (mathematics)|function]] {{<math|''>f''}}</math> is the set of [[ordered pair]]s {{<math|>(''x'', ''y'')}}</math>, where {{<math|''>f''(''x'') {{=}} ''y''}}.</math> In the common case where {{mvar|<math>x}}</math> and {{<math| ''>f''(''x'')}}</math> are [[real number|real numbers]]s, these pairs are [[Cartesian coordinates]] of points in a [[two-dimensionalplane space(geometry)|plane]] and thusoften form a subset[[Plane of this planecurve|curve]].

The graphical representation of the graph of a [[Function (mathematics)|function]] is also known as a ''[[Plot (graphics)|plot]]''.

In the case of [[Bivariate function|functions of two variables,]] – that is, functions whose [[Domain of a function|___domain]] consists of pairs {{<math|>(''x'', ''y'')}}</math> –, the graph usually refers to the set of [[ordered triple]]s {{<math|>(''x'', ''y'', ''z'')}}</math> where {{<math|''>f''(''x'', ''y'') {{=}} ''z''}}, instead of the pairs {{</math|((''x'', ''y''), ''z'')}} as in the definition above>. This set is a subset of [[three-dimensional space]]; for a continuous [[real-valued function]] of two real variables, itits isgraph forms a [[Surface (mathematics)|surface]], which can be visualized as a ''[[surface plot (graphics)|surface plot]]''.

{{anchor|graph of a relation}}

A graph of a function is a special case of a [[relation (mathematics)|relation]].

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.

{{anchor|graph of a relation}}A graph of a function is a special case of a [[relationRelation (mathematics)|relation]].

In the modern [[foundations of mathematics]], and, typically, in [[set theory]], a function is actually equal to its graph.<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|orig-year=1971|publisher=Dover Publications|isbn=978-0-486-79549-2|pages=49}}</ref> However, it is often useful to see functions as [[Map (mathematics)|mappings]],<ref>{{cite book|author=T. M. Apostol|authorlink=Tom M. Apostol|title=Mathematical Analysis|year=1981|publisher=Addison-Wesley|page=35}}</ref> which consist not only of the relation between input and output, but also which set is the ___domain, and which set is the [[codomain]]. For example, to say that a function is onto ([[Surjective function|surjective]]) or not the codomain should be taken into account. The graph of a function on its own doesn'tdoes not determine the codomain. It is common<ref>{{cite book|author=P. R. Halmos|title=A Hilbert Space Problem Book|url=https://archive.org/details/hilbertspaceprob00halm_811|url-access=limited|year=1982|publisher=Springer-Verlag|isbn=0-387-90685-1|page=[https://archive.org/details/hilbertspaceprob00halm_811/page/n47 31]}}</ref> to use both terms ''function'' and ''graph of a function'' since even if considered the same object, they indicate viewing it from a different perspective.

[[File:X^4 - 4^x.PNG|class=skin-invert-image|350px|thumb|Graph of the function {{nowrap|1=''<math>f''(''x'') = ''x''^{^4} −- 4<sup>''^x''</supmath>}} over the [[Interval (mathematics)|interval]] [−2,+3]. Also shown are the two real roots and the local minimum that are in the interval.]]

== Definition ==

Given a mapping[[function (mathematics)|function]] <math>f : X \to Y</math>, in other wordsfrom a functionset <math>f</math>{{mvar|X}} together(the with[[___domain itsof a function|___domain]]) <math>X</math>to anda codomainset <math>{{mvar|Y</math>}} (the [[codomain]]), the graph of the mappingfunction is the set<ref>{{cite book|author=D. S. Bridges|title=Foundations of Real and Abstract Analysis|url=https://archive.org/details/springer_10.1007-978-0-387-22620-0|year=1991|publisher=Springer|page=[https://archive.org/details/springer_10.1007-978-0-387-22620-0/page/n292 285]|isbn=0-387-98239-6}}</ref> the set

:<math display=block>G(f) = \{(x,f(x)) \mid: x \in X\},</math>,

which is a subset of the [[Cartesian product]] <math>X\times Y</math>. In the definition of a function in terms of [[set theory]], it is common to identify a function with its graph, although, formally, a function is formed by the triple consisting of its ___domain, its codomain and its graph.

which is a subset of <math>X\times Y</math>. In the abstract definition of a function, <math>G(f)</math> is actually equal to <math>f</math>.

One can observe that, if, <math>f:\mathbb R^n \to \mathbb R^m </math>, then the graph <math>G(f)

</math> is a subset of <math>\mathbb R^{n+m}</math> (strictly speaking it is <math>\mathbb R^n \times \mathbb R^m</math>, but one can embed it with the natural isomorphism).

== Examples ==

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=== Functions of one variable ===

[[File:Three-dimensional graph.png|right|thumb|250px|Graph of the [[functionFunction (mathematics)|function]] {{nowrap|1=''<math>f''(''x'', ''y'') = \sin\left(''x''^{^2}\right) ·\cdot \cos\left(''y''<sup>^2\right).</supmath>)}}.]]

The graph of the function <math>f : \{1,2,3\} \to \{a,b,c,d\}</math> defined by

: <math display=block>f(x)=

\begin{cases}

a, & \text{if }x=1, \\ d, & \text{if }x=2, \\ c, & \text{if }x=3,

\end{cases}

</math>

is the subset of the set <math>\{1,2,3\} \times \{a,b,c,d\}</math>

: <math display=block>G(f) = \{ (1,a), (2,d), (3,c) \}. \, </math>

From the graph, the ___domain <math>\{1,2,3\}</math> is recovered as the set of first component of each pair in the graph <math>\{1,2,3\} = \{x :\ \text{there exists } y,\text{ such that }(x,y) \in G(f)\}</math>.

Similarly, the [[Range of a function|range]] can be recovered as <math>\{a,c,d\} = \{y : \text{there exists }x,\text{ such that }(x,y)\in G(f)\}</math>.

The codomain <math>\{a,b,c,d\}</math>, however, cannot be determined from the graph alone.

The graph of the cubic polynomial on the [[real line]]

: <math display=block>f(x) = x^3 - 9x \, </math>

: <math>f(x) = x^3 - 9x \, </math>

is

: <math display=block> \{ (x, x^3 - 9x) : x \text{ is a real number} \}. \, </math>

: <math> \{ (x, x^3 - 9x) : x \text{ is a real number} \}. \, </math>

If this set is plotted on a [[Cartesian plane]], the result is a curve (see figure).

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=== Functions of two variables ===

[[File:F(x,y)=−((cosx)^2 + (cosy)^2)^2.PNG|class=skin-invert-image|thumb|250px|Plot of the graph of {{nowrap|1=''<math>f''(''x'', ''y'') = −- \left(\cos\left(''x''^{^2}\right) + \cos\left(''y''^{^2}\right)\right)<sup>^2,</supmath>}}, also showing its gradient projected on the bottom plane.]]

The graph of the [[trigonometric function]]

: <math display=block> f(x,y) = \sin(x^2)\cos(y^2) \, </math>

: <math> f(x,y) = \sin(x^2)\cos(y^2) \, </math>

is

: <math display=block> \{ (x, y, \sin(x^2) \cos(y^2)) : x \text{ and } y \text{ are real numbers} \}. </math>

: <math> \{ (x, y, \sin(x^2) \cos(y^2)) : x \text{ and } y \text{ are real numbers} \}. </math>

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:

: <math display=block> f(x, y) = -(\cos(x^2) + \cos(y^2))^2 \, .</math>

: <math> f(x,y) = -(\cos(x^2) + \cos(y^2))^2 \, </math>

== 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 are not 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]].

== See also ==

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* [[Asymptote]]

* [[Chart]]

* [[Plot (graphics)|Plot]]

* [[Concave function]]

* [[Convex function]]

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== References ==

{{reflist}}

== Further reading ==

{{refbegin}}

* {{Zălinescu Convex Analysis in General Vector Spaces 2002}} <!-- {{sfn|Zălinescu|2002|pp=}} -->

{{refend}}

== External links ==

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{{Calculus topics}}

{{Visualization}}

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[[Category:Charts]]