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.
Definition
Given a
function from a set X (the
domain) to a set Y (the
codomain), the graph of the function is the set[4]
which is a subset of the
Cartesian product. In the definition of a function in terms of
set theory, it is common to identify a function with its graph, although, formally, a function is formed by the triple consisting of its domain, its codomain and its graph.
Examples
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
is
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: