Chapter 2.4, Problem 41AYU

In Problems 43-46, the graph of a piecewise-defined function is given. Write a definition for each function.

To determine

To find:The definition of the given piecewise-defined function.

Answer to Problem 41AYU

Solution:

The required definition is $f\left(x\right)=\{\begin{array}{c}-x if -1\le x<0 \\ \frac{x}{2} if 0<x\le 2\end{array}$.

Explanation of Solution

Given:

Graph is given as follows:

It is asked us to find the definition.

There are two line segments in the given graph. One is from the point $\left(-1,1\right)\text{ to }\left(0,0\right)$ and other is from $\left(0,0\right)\text{ to }\left(2,1\right)$. The first line segment lies from 1 to 0 and second line segment lies from 0 to 2.

Formula used:

Equation of a line when two points are given, $\frac{y-y_1}{y_2-y_1}=\frac{x-x_1}{x_2-x_1}$

For the points $\left(-1,1\right)\text{ to }\left(0,0\right)$ ,

$\frac{y-1}{0-1}=\frac{x+1}{0+1}$

$\frac{y-1}{-1}=\frac{x+1}{1}$

$y-1=-x-1$

$y=-x$

With the condition, $-1\le x<0$.

For the points $\left(0,0\right)\text{ to }\left(2,1\right)$ ,

$\frac{y-0}{1-0}=\frac{x-0}{2-0}$

$\frac{y}{1}=\frac{x}{2}$

$y=\frac{x}{2}$

With the condition, $0<x\le 2$.

The required definition is $f\left(x\right)=\{\begin{array}{c}-x if -1\le x<0 \\ \frac{x}{2} if 0<x\le 2\end{array}$.