Graph of a function

In mathematics, the graph of a function $f$ is the set of ordered pairs $(x,y)$ , where $f(x)=y.$ In the common case where $x$ and $f(x)$ are real numbers, these pairs are Cartesian coordinates of points in a plane and often form a curve. The graphical representation of the graph of a function is also known as a plot.

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

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 axes; see Plot (graphics) for details.

A graph of a function is a special case of a relation. In the modern foundations of mathematics, and, typically, in set theory, a function is actually equal to its graph.^[1] However, it is often useful to see functions as mappings,^[2] 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) 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 terms function and graph of a function since even if considered the same object, they indicate viewing it from a different perspective.

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

[Pinter2014-1] Charles C Pinter (2014) [1971]. A Book of Set Theory. Dover Publications. p. 49. ISBN 978-0-486-79549-2.

[2] T. M. Apostol (1981). Mathematical Analysis. Addison-Wesley. p. 35.

[3] P. R. Halmos (1982). A Hilbert Space Problem Book. Springer-Verlag. p. 31. ISBN 0-387-90685-1.

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Function
$x \mapsto f (x)$
History of the function concept
Types by domain and codomain
$X \to 𝔹$ $𝔹 \to X$ $𝔹 n \to X$ $X \to ℤ$ $ℤ \to X$ $X \to ℝ$ $ℝ \to X$ $ℝ n \to X$ $X \to ℂ$ $ℂ \to X$ $ℂ n \to X$
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