Given a mapping in other words a function together with its domain and codomain the graph of the mapping is[4] the set

which is a subset of . In the abstract definition of a function, is actually equal to

One can observe that, if, then the graph is a subset of (strictly speaking it is but one can embed it with the natural isomorphism).


Functions of one variable

Graph of the function

The graph of the function defined by

is the subset of the set

From the graph, the domain is recovered as the set of first component of each pair in the graph . Similarly, the range can be recovered as . The codomain , however, cannot be determined from the graph alone.

The graph of the cubic polynomial on the real line


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

Functions of two variables

Plot of the graph of also showing its gradient projected on the bottom plane.

The graph of the trigonometric function


If this set is plotted on a 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:

See also


  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.
  4. D. S. Bridges (1991). Foundations of Real and Abstract Analysis. Springer. p. 285. ISBN 0-387-98239-6.

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