In mathematics, the graph of a function is the collection of all ordered pairs
. If the
function input
is an ordered pair
of real numbers, the graph is the collection of all ordered
triples
, and for a continuous function is a surface (see three-dimensional graph).
Informally, if is a real number and
is a real function, graph may mean the graphical representation
of this collection, in the form of a line chart: a curve on a Cartesian plane, together with Cartesian axes, etc.
Graphing on a Cartesian plane is sometimes referred to as curve sketching. The graph of a function on real numbers may
be mapped directly to the graphic representation of the function. For general functions, a graphic representation cannot
necessarily be found and the formal definition of the graph of a function suits the need of mathematical statements, e.g.,
the closed graph theorem in functional analysis.
The concept of the graph of a function is generalized to the graph of a relation. Note that although a function is always identified with its graph, they are not the same because it will happen that two functions with different codomain could have the same graph. For example, the cubic polynomial mentioned below is a surjection if its codomain is the real numbers but it is not if its codomain is the complex field.
To test whether a graph of a curve is a function of , use the vertical line test. To test whether a graph of a
curve is a function of
, use the horizontal line test. If the function has an inverse, the graph of the inverse
can be found by reflecting the graph of the original function over the line
.