Question

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

Short answer

The piecewise-defined function is,

$f\left(x\right)=\left\{\begin{array}{l}-x-1\le x\le 0\\ \frac{1}{2}x0\le x\le 2\end{array}\right.$.

Step-by-step solution

Step 1. Given Information

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

Step 2. Defining the function

It can be seen from the graph that $(0,0)$is a point on the graph, So when $x=0$,

$f\left(0\right)=0$.

Similarly, $(-1,1)$is a point on the graph, So when $x=-1$,

$f(-1)=1$

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

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

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

Step 3. Defining the function

For the point $(2,1)$on the graph, whenlocalid="1645867197330" $x=2$,

$f\left(2\right)=1$

From the graph, we can see that for the points with the $x$-coordinates between $0$and $2$, both inclusive, the value of the function is half of the value of the $x$-coordinate.

So, $f\left(2\right)=\frac{2}{2}=1$

So, the function will be,

$f\left(x\right)=\left\{\begin{array}{l}-x-1\le x\le 0\\ \frac{1}{2}x0\le x\le 2\end{array}\right.$