# Graphic determination of the domain and of the image

To determine the domain and the path of a function from its graph, we will concentrate on all the represented pairs of numbers $(x, y)$.

• A real number $x = a$ belongs to the domain of a function if and only if the vertical straight line $x = a$ is in the graph of the function at some point.
• A real number $y = b$ belongs to the image of a function if and only if the horizontal straight line $y = b$ cuts the graph of the function at some point.

Determine the domain and the image of the following function $f$ defined by parts: We observe that the graph of the function is not continuous. To the left of $0$ the function is a straight line with slope equal to $-1$.

At $x=0$ the function takes the value $1$. While when $x$ is greater than zero but smaller than $2$, the slope is $1$.

Finally when $x$ is greater than $3$ the slope is $0$, and $y$ always takes the value $1$.

This way, the domain will be the set of the real numbers except fort the part in which the function is not defined, which is given by the interval $[2, 3)$.

Therefore, $Dom (f) = \mathbb{R}-[2,3)=(-\infty,2) \cup [3,+\infty)$.

On the other hand we can realize that the path of the function is the set of the real ones $x> 0$.

Then, $Im (f) = (0,+\infty)=\mathbb{R}^+$

Finally, we present the analytical expression of the function:

$$f(x)=\left\{ \begin{array}{rcl} -x & \mbox{ if } & x < 0 \\ 1 & \mbox{if} & x=0 \\ x & \mbox{if} & 0< x < 2 \\ 1 &\mbox{if} & x\geq 3 \end{array} \right.$$