Graph of a function

{{ safesubst:#invoke:Unsubst||$N=Refimprove |date=__DATE__ |$B= {{#invoke:Message box|ambox}} }} Template:Mergefrom {{#invoke:Hatnote|hatnote}} Graph of the the function f(x) = x4 − 4x over the interval [−2,+3]. Also shown are its two real roots and global minimum over the same interval.

In mathematics, the graph of a function f is the collection of all ordered pairs (x, f(x)). If the function input x is a scalar, the graph is a two-dimensional graph, and for a continuous function is a curve. If the function input x is an ordered pair (x1, x2) of real numbers, the graph is the collection of all ordered triples (x1, x2, f(x1, x2)), and for a continuous function is a surface (see three-dimensional graph).

Informally, if x is a real number and f 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 x, use the vertical line test. To test whether a graph of a curve is a function of y, 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 y = x.

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 foundation of mathematics known as set theory, a function and its graph are essentially the same thing.

Examples

Functions of one variable

The graph of the function.

$f(x)=\left\{{\begin{matrix}a,&{\mbox{if }}x=1\\d,&{\mbox{if }}x=2\\c,&{\mbox{if }}x=3.\end{matrix}}\right.$ is

{ (1,a), (2,d), (3,c) }.

The graph of the cubic polynomial on the real line

$f(x)=x^{3}-9x$ is

{ (x, x3 − 9x) : x is a real number }.

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 on the real line

f(x, y) = sin(x2) · cos(y2)

is

{ (x, y, sin(x2) · cos(y2)) : x and y are real numbers }.

If this set is plotted on a three dimensional Cartesian coordinate system, the result is a surface (see figure).

Often times 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 graph of the function:

f(x, y) = −(cos(x2) + cos(y2))2

Normal to a graph

Given a function f of n variables: $x=x_{1},\dotsc ,x_{n}$ , the normal to the graph is

$(\nabla f,-1)$ (up to multiplication by a constant). This is seen by considering the graph as a level set of the function $g(x,z)=f(x)-z$ , and using that $\nabla g$ is normal to the level sets.

Generalizations

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 bundles aren't cartesian products, but appear to be up close. There is a corresponding notion of a graph on a fibre bundle called a section.