Graph of A Function
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Graph of A Function
Graph of the function

In mathematics, the graph of a function f is, formally, the set of all ordered pairs (x, f(x)), such that x is in the domain of the function f. In the common case where x and f(x) are real numbers, these pairs are Cartesian coordinates of points in the Euclidean plane and thus form a subset of this plane, which is a curve in the case of a continuous function. This graphical representation of the function is also called the graph of the function.

In the case of functions of two variables, that is functions whose domain consists of pairs (x, y), the graph can be identified with the set of all ordered triples (x, y, f(x, y)). For a continuous real-valued function of two real variables, the graph 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 in 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 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 doesn't 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.

Graph of the function over the interval [-2,+3]. Also shown are the two real roots and the local minimum that are in the interval.

## Definition

Given a mapping ${\displaystyle f:X\to Y}$, in other words a function ${\displaystyle f}$ together with its domain ${\displaystyle X}$ and codomain ${\displaystyle Y}$, the graph of the mapping is[4] the set

${\displaystyle G(f)=\{(x,f(x))\mid x\in X\}}$,

which is a subset of ${\displaystyle X\times Y}$. In the abstract definition of a function, ${\displaystyle G(f)}$ is actually equal to ${\displaystyle f}$.

## Examples

### Functions of one variable

Graph of the function .

The graph of the function ${\displaystyle f:\{1,2,3\}\mapsto \{a,b,c,d\}}$ defined by

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

is the subset of the set ${\displaystyle \{1,2,3\}\times \{a,b,c,d\}}$

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

From the graph, the domain ${\displaystyle \{1,2,3\}}$ is recovered as the set of first component of each pair in the graph ${\displaystyle \{1,2,3\}=\{x:\ {\text{there exists }}y,{\text{ such that }}(x,y)\in G(f)\}}$. Similarly, the range can be recovered as ${\displaystyle \{a,c,d\}=\{y:{\text{there exists }}x,{\text{ such that }}(x,y)\in G(f)\}}$. The codomain ${\displaystyle \{a,b,c,d\}}$, however, cannot be determined from the graph alone.

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 , 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:

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

## 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.