, the graph of a function
is the set of ordered pair
s , where . In the common case where and 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 , the graph usually refers to the set of ordered triple
s where , instead of the pairs 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
, 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.
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 doesn't 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
, the graph of the mapping is
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
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
, 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).
Functions of two variables
The graph of the trigonometric function
If this set is plotted on a three dimensional Cartesian coordinate system
, the result is a surface (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:
The graph of a function is contained in a Cartesian product
of sets. An X–Y plane is a cartesian product of two lines, called X and Y, while a cylinder is a cartesian product of a line and a circle, whose height, radius, and angle assign precise locations of the points. Fibre bundle
s are not Cartesian products, but appear to be up close. There is a corresponding notion of a graph on a fibre bundle called a section
* Concave function
* Convex function
* Contour plot
* Critical point
* Normal to a graph
* Stationary point
* Vertical translation
* Weisstein, Eric W.Function Graph
" From MathWorld—A Wolfram Web Resource.
Category:Functions and mappings
Category:Numerical function drawing