# Graph_of_a_function

## Definition

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

$G(f)=\{(x,f(x)):x\in X\},$ which is a subset of $X\times Y$ . In the abstract definition of a function, $G(f)$ is actually equal to $f.$ One can observe that, if, $f:\mathbb {R} ^{n}\to \mathbb {R} ^{m},$ then the graph $G(f)$ is a subset of $\mathbb {R} ^{n+m}$ (strictly speaking it is $\mathbb {R} ^{n}\times \mathbb {R} ^{m},$ but one can embed it with the natural isomorphism).

## Examples

#### Functions of one variable Graph of the function f ( x , y ) = sin ⁡ ( x 2 ) ⋅ cos ⁡ ( y 2 ) . {\displaystyle f(x,y)=\sin \left(x^{2}\right)\cdot \cos \left(y^{2}\right).}

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

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

The graph of the cubic polynomial on the real line

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

$\{(x,x^{3}-9x):x{\text{ is a real number}}\}.$ 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 ⁡ ( x 2 ) + cos ⁡ ( y 2 ) ) 2 , {\displaystyle f(x,y)=-\left(\cos \left(x^{2}\right)+\cos \left(y^{2}\right)\right)^{2},} also showing its gradient projected on the bottom plane.

The graph of the trigonometric function

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

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

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