Problems from Graphic determination of the domain and of the image

Development:

A way of proceeding is to draw the graph of the function and then find the domain and the image.

We may realize that:

• In the interval $(-\mathrm{\infty },0]$ we have a straight line of slope $m=1$ and that cuts the axis $x$ in $x=-2$.

• In the interval $(0,2]$, we have a constant function $y=2$.

• In the interval $(2,+\mathrm{\infty })$ we have a straight line of slope $m=-1$ and that cuts the axis $x$ in $x=4$.

Therefore the graph of the function is:

Thus it is clear that the domain of the function is:

$Dom(f)=(-\mathrm{\infty },+\mathrm{\infty })$

and that its image is:

$Im(f)=(-\mathrm{\infty },2]$

Solution:

$Dom(f)=(-\mathrm{\infty },+\mathrm{\infty })$

$Im(f)=(-\mathrm{\infty },2]$

Hide solution and development