# Graph of a function

In mathematics, the graph of a function ${\displaystyle f}$ is the set of ordered pairs ${\displaystyle (x,y)}$, where ${\displaystyle f(x)=y.}$ In the common case where ${\displaystyle x}$ and ${\displaystyle f(x)}$ 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 ${\displaystyle (x,y),}$ the graph usually refers to the set of ordered triples ${\displaystyle (x,y,z)}$ where ${\displaystyle f(x,y)=z,}$ instead of the pairs ${\displaystyle ((x,y),z)}$ 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.