Chapter 0.2, Problem 46E

To determine

To find: The piece wise formula for the function.

Answer to Problem 46E

  $f\left(x\right)=\left\{\begin{array}{ccc}\frac{1}{2}x& ,\text{ if}& -2\le x\le 0\\ -2x+2& ,\text{ if}& 0<x\le 1\\ -1& ,\text{ if}& 1<x\le -1\end{array}\right.$

Explanation of Solution

Given information:

The given graph of the piece wise function is

  

Concept used:

Two point form equation of line:

  $y-y_1=\frac{y_2-y_1}{x_2-x_1}\left(x-x_1\right)$

Calculations:

There three piece of graph. So, split graph into three interval.

Open circle means the value of $x$ not included and closed circle means the value of $x$ included.

First piece of the graph, $-2\le x\le 0$ .

The equation of line is: $y=\frac{1}{2}x$

Second piece of the graph, $0<x\le 1$

The equation of line: $y=-2x+2$

Third piece of the graph, $1<x\le 3$

The equation of the line: $y=-1$

Thus, the piecewise function is,

  $f\left(x\right)=\left\{\begin{array}{ccc}\frac{1}{2}x& ,\text{ if}& -2\le x\le 0\\ -2x+2& ,\text{ if}& 0<x\le 1\\ -1& ,\text{ if}& 1<x\le -1\end{array}\right.$