Graph of a function: Difference between revisions

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

{{refimprove|date=August 2014}}

[[File:F(x) = x^3 − 9x.PNG|right|thumb|250px| Graph of the function {{nowrap|1=''<math>f''(''x'') = ''x''<sup>^3 - 9x.</supmath> − 9''x''}}]]

In [[mathematics]], the '''graph''' of a [[functionFunction (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|real numbers]]s, these pairs are [[Cartesian coordinates]] of points in [[two-dimensional space]] and thus form a subset of this plane.

In the case of 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, it is a [[Surface (mathematics)|surface]].

{{anchor|graph of a relation}}

A graph of a function is a special case of a [[relationRelation (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.

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|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't 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|350px|thumb|Graph of the function {{nowrap|1=''<math>f''(''x'') = ''x''^{^4} −- 4<sup>''x''4x</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 <math>f : X \to Y,</math>, in other words a function <math>f</math> together with its ___domain <math>X</math> and codomain <math>Y,</math>, the graph of the mapping is<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 <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>\R^{n+m}</math> (strictly speaking it is <math>\R^n \times \R^m,</math> but one can embed it with the natural isomorphism).

</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|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<math>X</math>–<math>Y</math> plane is a cartesianCartesian product of two lines, called <math>X</math> and <math>Y,</math> 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]].

The {{visible anchor|graph of a multifunction}}, say the [[multifunction]], <math>\mathcal{R} : X \rightrightarrows Y</math> is the set <math>\operatorname{gr} \mathcal{R} := \left\{ (x, y) \in X \times Y : y \in \mathcal{R}(x) \right\}.</math>

== See also ==

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== References ==

{{reflist}}

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

== External links ==