# Graph of a function

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Graph of 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, formally, the set of all ordered pairs (x, f(x)), and, in practice, the graphical representation of this set. If the function input x is a real number, 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.

The concept of the graph of a function is generalized to the graph of a relation. To test whether a graph of a relation represents a function of the first variable x, one uses the vertical line test. To test whether a graph represents a function of the second variable y, one uses 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 foundations of mathematics, and, typically, in set theory, a function and its graph are the same thing.[1] However, the concept of function is aimed to formalize the intuitive notion of "mapping", which is not directly apparent on the graph. Therefore, reasoning about functions is generally easier if a function is distinguished from its graphs, and if the graph is viewed as a graphical representation of the function.

Graph of the function f(x) = x3 − 9x

## Examples

### Functions of one variable

Graph of the function f(x, y) = sin(x2) · cos(y2).

The graph of the function.

${\displaystyle f(x)={\begin{cases}a,&{\text{if }}x=1,\\d,&{\text{if }}x=2,\\c,&{\text{if }}x=3,\end{cases}}}$

is

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

The graph of the cubic polynomial on the real line

${\displaystyle f(x)=x^{3}-9x\,}$

is

${\displaystyle \{(x,x^{3}-9x):x{\text{ is a real number}}\}.\,}$

If this set is plotted on a Cartesian plane, the result is a curve (see figure).

### Functions of two variables

Plot of the graph of f(x, y) = −(cos(x2) + cos(y2))2, also showing its gradient projected on the bottom plane.

The graph of the trigonometric function

${\displaystyle f(x,y)=\sin(x^{2})\cos(y^{2})\,}$

is

${\displaystyle \{(x,y,\sin(x^{2})\cos(y^{2})):x{\text{ and }}y{\text{ are real numbers}}\}.}$

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:

${\displaystyle f(x,y)=-(\cos(x^{2})+\cos(y^{2}))^{2}\,}$

### Normal to a graph

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

${\displaystyle (\nabla f,-1)}$

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