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. However, it is often useful to see functions as mappings, 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 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 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
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).
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: