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 two-dimensional space and thus form a subset of this plane.

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*, instead of the pairs ((*x*, *y*), *z*) as in the definition above. This set is a subset of three-dimensional space; for a continuous real-valued function of two real variables, it is a surface.

A graph of a function is a special case of a relation.

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.

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 doesn't 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.

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

A graph of a function is a special case of a relation.

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.

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 doesn't 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.

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, . 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).

The graph of the function defined by