# Problems from Graphic determination of the domain and of the image

Consider the following function defined in parts:

$$f(x)=\Bigg\lbrace \begin{eqnarray} x+2 & \mbox{si} & x\leq 0 \\\\ 2 & \mbox{si} & 0 < x \leq 2 \\\\ -x+4 & \mbox{si} & x>2 \end{eqnarray}$$

Do the graphic representation. Find the domain and the image of the function.

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### 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 $$(-\infty, 0]$$ 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 $$(2, +\infty)$$ 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:

$$Dom (f) = (-\infty, +\infty)$$

and that its image is:

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

### Solution:

$$Dom (f) = (-\infty, +\infty)$$

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

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