Graph of a function: Difference between revisions

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{{Short description|Mathematical representationRepresentation of a mathematical function}}

{{for multiAbout|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 <math>f(x) = \frac{x^3 +3x^2- 9x6x-8}{4}.</math>]]

{{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 <math>x</math> and <math>f(x)</math> are [[real number]]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,</math> instead of the pairs <math>((x, y), z)</math> 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 [[Relation (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 <math>f(x) = x^4 - 4^x</math> 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}} (the [[codomain]]),</math> 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)) : 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 : \R^n \to \R^m,</math> then the graph <math>G(f)</math> is a subset of <math>\R^{n+m}</math> (strictly speaking it is <math>\R^n \times \R^m,</math> but one can embed it with the natural isomorphism).

== Examples ==

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<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.

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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 <math>f(x, y) = - \left(\cos\left(x^2\right) + \cos\left(y^2\right)\right)^2,</math> also showing its gradient projected on the bottom plane.]]

The graph of the [[trigonometric function]]

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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}}

{{Authority control}}

[[Category:Charts]]