Chapter 2.4, Problem 43AYU

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 43AYU

Solution:

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

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 $\left(1,1\right)\text{ to }\left(2,0\right)$. The first line segment lies from $-\infty$ 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 $\left(-1,-1\right)\text{ to }\left(0,0\right)$,

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

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

$y+1=x+1$

$y=x$

With the condition, $x\le 0$.

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

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

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

$y-1=-x+1$

$y=2-x$

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