Graph of a function: Difference between revisions

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{{Short description|Representation of a mathematical function}}

{{for|the graph-theoretic representation of a function from a set to the same set|Functional graph}}

{{About||graph-theoretic representation of a function|Functional graph}}

[[File:X^4-4^x.gif|2900px|thumbnail|right|Graph of the function {{nowrap|1=''f''(''x'') = ''x''⁴ − 4^''x''}} {{-}}(−2, +2)]]

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

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''₁, ''x''₂)}} of real numbers, the graph is the collection of all [[ordered triple]]s {{nowrap|(''x''₁, ''x''₂, ''f''(''x''₁, ''x''₂))}}, and for a [[continuous function]] is a [[surface]] (see [[three-dimensional graph]]).

[[File:Polynomial of degree three.svg|class=skin-invert-image|thumb|250x250px|Graph of the function <math>f(x)=\frac{x^3+3x^2-6x-8}{4}.</math>]]

{{functions}}

In [[mathematics]], the '''graph of a 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 [[plane (geometry)|plane]] and often form a [[Plane curve|curve]].

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 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>. This is a subset of [[three-dimensional space]]; for a continuous [[real-valued function]] of two real variables, its graph forms a [[Surface (mathematics)|surface]], which can be visualized as a ''[[surface plot (graphics)|surface plot]]''.

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.

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

== Examples ==

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 does 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.

[[Image:cubicpoly.png||right|thumb|300 px| Graph of the function {{nowrap|1=''f''(''x'') = ''x''³ − 9''x''}}]]

[[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 ==

=== Functions of two variables ===

Given a [[function (mathematics)|function]] <math>f : X \to Y</math> from a set {{mvar|X}} (the [[___domain of a function|___domain]]) to a set {{mvar|Y}} (the [[codomain]]), the graph of the function 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 graph of the [[trigonometric]] function on the real line

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

: ''f''(''x'', ''y'') = [[sine|sin]](''x''²) · [[cosine|cos]](''y''²)

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.

is

: {(''x'', ''y'', sin(''x''²) · cos(''y''²)) : ''x'' and ''y'' are real numbers}.

== Examples ==

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'')² + sin(''y'')²}}

</gallery>

[[File:Three-dimensional graph.png|right|thumb|250px|Graph of the [[Function (mathematics)|function]] <math>f(x, y) = \sin\left(x^2\right) \cdot \cos\left(y^2\right).</math>]]

=== 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.

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

== Generalizations ==

<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 :\ \exists y,\text{ such that }(x,y) \in G(f)\}</math>.

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]].

Similarly, the [[Range of a function|range]] can be recovered as <math>\{a,c,d\} = \{y : \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]]

== Tools for plotting function graphs ==

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

is

<math display=block>\{ (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).

=== Hardware ===

{{clear}}

=== Functions of two variables ===

* [[Graphing calculator]]

* [[Oscilloscope]]

* [[Paper]] and [[pencil]]

[[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.]]

=== Software ===

See [[List of graphing software]]

The graph of the [[trigonometric function]]

<math display=block>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>

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>

== See also ==

{{div col|colwidth=25em}}

<div style="-moz-column-count:2; column-count:2;">

* [[Asymptote]]

* [[Chart]]

* [[Plot (graphics)|Plot]]

* [[Concave function]]

* [[Convex function]]

* [[Contour line|Contour plot]]

* [[Critical point (mathematics)|Critical point]]

* [[Derivative]]

* [[Epigraph (mathematics)|Epigraph]]

* [[Normal (geometry)|Normal to a graph]]

* [[Chart]]

* [[Slope]]

* [[Stationary point]]

* [[Slope]]

* [[Solution point]]

* [[Tetraview]]

* [[Vertical translation]]

* [[Yy-intercept]]

{{div col end}}

* [[Graph theory]]

</div>

== References ==

{{reflist}}

== Further reading ==

{{refbegin}}

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

{{refend}}

== External links ==

{{Commons category|GraphsFunction plots}}

* [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.

{{Calculus topics}}

{{Visualization}}

{{Authority control}}

[[Category:Charts]]

[[Category:Functions and mappings]]

[[Category:Numerical function drawing]]

[[pt:Função#Gráficos de função]]