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Graph of a function

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Representation of a mathematical function

For graph-theoretic interpretation of a function, seeFunctional graph.

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Graph of the function $f(x)={\frac {x^{3}+3x^{2}-6x-8}{4}}.$

{\displaystyle f(x)={\frac {x^{3}+3x^{2}-6x-8}{4}}.} — Graph of the function $f(x)={\frac {x^{3}+3x^{2}-6x-8}{4}}.$

Function
x ↦f (x)
History of the function concept
Types bydomain andcodomain
`X` →𝔹 𝔹 →`X` 𝔹ⁿ →`X` `X` →ℤ ℤ →`X` `X` →ℝ ℝ →`X` ℝⁿ →`X` `X` →ℂ ℂ →`X` ℂⁿ →`X`
Classes/properties
Constant Identity Linear Polynomial Rational Algebraic Analytic Smooth Continuous Measurable Injective Surjective Bijective
Constructions
Restriction Composition λ Inverse
Generalizations
Relation (Binary relation) Set-valued Multivalued Partial Implicit Space Higher-order Morphism Functor
List of specific functions
v t e

Inmathematics, thegraph of a function $f {\displaystyle f}$ is the set ofordered pairs $(x,y)$ , where $f(x)=y.$ In the common case where $x {\displaystyle x}$ and $f(x)$ arereal numbers, these pairs areCartesian coordinates of points in aplane and often form acurve.The graphical representation of the graph of afunction is also known as aplot.

In the case offunctions of two variables – that is, functions whosedomain consists of pairs $(x,y)$ –, the graph usually refers to the set ofordered triples $(x,y,z)$ where $f(x,y)=z$ . This is a subset ofthree-dimensional space; for a continuousreal-valued function of two real variables, its graph forms asurface, which can be visualized as asurface plot.

Inscience,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 usingrectangular axes; seePlot (graphics) for details.

A graph of a function is a special case of arelation. In the modernfoundations of mathematics, and, typically, inset theory, a function is actually equal to its graph.^[1] However, it is often useful to see functions asmappings,^[2] which consist not only of the relation between input and output, but also which set is the domain, and which set is thecodomain. For example, to say that a function is onto (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^[3] to use both termsfunction andgraph of a function since even if considered the same object, they indicate viewing it from a different perspective.

Definition

Given afunction $f:X\to Y$ from a setX (thedomain) to a setY (thecodomain), the graph of the function is the set^[4] $G(f)=\{(x,f(x)):x\in X\},$ which is a subset of theCartesian product $X\times Y$ . In the definition of a function in terms ofset 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.

Examples

Functions of one variable

Graph of the function $f(x)=x^{4}-4^{x}$ over theinterval [−2,+3]. Also shown are the two real roots and the local minimum that are in the interval.

{\displaystyle f(x)=x^{4}-4^{x}} — Graph of the function $f(x)=x^{4}-4^{x}$ over theinterval [−2,+3]. Also shown are the two real roots and the local minimum that are in the interval.

The graph of the function $f:\{1,2,3\}\to \{a,b,c,d\}$ defined by $f(x)={\begin{cases}a,&{\text{if }}x=1,\\d,&{\text{if }}x=2,\\c,&{\text{if }}x=3,\end{cases}}$ is the subset of the set $\{1,2,3\}\times \{a,b,c,d\}$ $G(f)=\{(1,a),(2,d),(3,c)\}.$

From the graph, the domain $\{1,2,3\}$ is recovered as the set of first component of each pair in the graph $\{1,2,3\}=\{x:\ \exists y,{\text{ such that }}(x,y)\in G(f)\}$ .Similarly, therange can be recovered as $\{a,c,d\}=\{y:\exists x,{\text{ such that }}(x,y)\in G(f)\}$ .The codomain $\{a,b,c,d\}$ , however, cannot be determined from the graph alone.

The graph of the cubic polynomial on thereal line $f(x)=x^{3}-9x$ is $\{(x,x^{3}-9x):x{\text{ is a real number}}\}.$

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

Functions of two variables

Further information:Bivariate surface

Plot of the graph of $f(x,y)=-\left(\cos \left(x^{2}\right)+\cos \left(y^{2}\right)\right)^{2},$ also showing its gradient projected on the bottom plane.

{\displaystyle f(x,y)=-\left(\cos \left(x^{2}\right)+\cos \left(y^{2}\right)\right)^{2},} — Plot of the graph of $f(x,y)=-\left(\cos \left(x^{2}\right)+\cos \left(y^{2}\right)\right)^{2},$ also showing its gradient projected on the bottom plane.

The graph of thetrigonometric function $f(x,y)=\sin(x^{2})\cos(y^{2})$ is $\{(x,y,\sin(x^{2})\cos(y^{2})):x{\text{ and }}y{\text{ are real numbers}}\}.$

If this set is plotted on athree 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: $f(x,y)=-(\cos(x^{2})+\cos(y^{2}))^{2}.$

See also

References

^Pinter, Charles C. (2014) [1971].A Book of Set Theory. Dover Publications. p. 49.ISBN 978-0-486-79549-2.
^Apostol, T. M. (1981).Mathematical Analysis. Addison-Wesley. p. 35.
^Halmos, P. R. (1982).A Hilbert Space Problem Book. Springer-Verlag. p. 31.ISBN 0-387-90685-1.
^Bridges, D. S. (1991).Foundations of Real and Abstract Analysis. Springer. p. 285.ISBN 0-387-98239-6.

Further reading

Zălinescu, Constantin (30 July 2002).Convex Analysis in General Vector Spaces. River Edge, N.J. London:World Scientific Publishing.ISBN 978-981-4488-15-0.MR 1921556.OCLC 285163112 – viaInternet Archive.

External links

Wikimedia Commons has media related toFunction plots.

Weisstein, Eric W. "Function Graph." From MathWorld—A Wolfram Web Resource.

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