In Problems 41–44, the graph of a piecewise-defined function is given. Write a definition for each function.

We are given a graph of a piecewise-defined function,

It can be seen from the graph that the point $(0,0)$on the graph.

So, when $x=0$, $f\left(0\right)=0$.

Similarly, $(-1,1)$ is a point on the graph means that the y-coordinate $1$is the value of $f$at the $x$coordinate $-1$.

So, when $x=-1$,

$f(-1)=1$

It can be seen from the graph that for the points with the x-coordinates is between $0$and $-1$, both inclusive, the value of the y- coordinate is opposite of the value of the x-coordinate.

So,

$$\begin{gathered} f(-1)=-(-1) \\ =1 \end{gathered}$$.

So, the value of the function $f\left(x\right)$, when $-1\le x\le 0$is given by $-x$.

It can be seen from the graph that for the points with the x-coordinates between $0$and $2$, with $2$ inclusive, the value of the y-coordinate is $-x+2$

$$\begin{gathered} f\left(1\right)=-1+2 \\ =1 \end{gathered}$$

So, the value of the function when $0<x\le 2$is given by role="math" localid="1645873142560" $-x+2$.

So, the function will be,

$f\left(x\right)=\left\{\begin{array}{c}-xforx\le 0\\ -x+2for0<x\le 2\end{array}\right.$