Graphic determination of the domain and of the image

Example

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

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

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

Finally, we present the analytical expression of the function:

$$f(x)=\{\begin{array}{rcl}-x& \text{ if }& x<0\\ 1& \text{if}& x=0\\ x& \text{if}& 0<x<2\\ 1& \text{if}& x\ge 3\end{array}$$