Graph of a function information

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.

Graph of a function information

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Graph of a function

Cubic function

Implicit function theorem

Domain of a function

Periodic function

Graph

Convex function

Asymptote

Linear function

Inflection point

Differential calculus

Second derivative

Saddle point

Stationary point

Derivative

Domain coloring

Injective function

Weierstrass function

Linear equation

Vertical line test

Piecewise linear function

Lipschitz continuity

Limit of a function

Zero of a function

Transfer function

Surjective function

Curvature

Function
$x \mapsto f (x)$
History of the function concept
Examples of domains and codomains
$X$ → $\mathbb {B}$ , $\mathbb {B}$ → $X$ , $\mathbb {B} ^{n}$ → $X$ $X$ → $\mathbb {Z}$ , $\mathbb {Z}$ → $X$ $X$ → $\mathbb {R}$ , $\mathbb {R}$ → $X$ , $\mathbb {R} ^{n}$ → $X$ $X$ → $\mathbb {C}$ , $\mathbb {C}$ → $X$ , $\mathbb {C} ^{n}$ → $X$
Classes/properties
Constant Identity Linear Polynomial Rational Algebraic Analytic Smooth Continuous Measurable Injective Surjective Bijective
Constructions
Restriction Composition λ Inverse
Generalizations
Binary relation Partial Multivalued Implicit Space
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