Chapter 0.2, Problem 45E

To determine

To Write:

A piecewise-defined function for the given graph.

Answer to Problem 45E

  $f\left(x\right)=\left\{\begin{array}{ll}-x,\hfill & -1\le x<0\hfill \\ 1,\hfill & 0<x\le 1\hfill \\ \frac{3}{2}-\frac{x}{2},\hfill & 1<x<3\hfill \end{array}\right.$ .

Explanation of Solution

Given:

The graph:

  

Concepts Used:

Reading the graph of a piecewise function.

The equation of the graph of a line passing through the points $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$ is $\frac{y-y_1}{x-x_1}=\frac{y_2-y_1}{x_2-x_1}$ .

Isolating a variable from and equation.

The equation of the graph of a horizontal line at a distance $a$ units above the $x$ axis is $y=a$ .

Calculations:

The given graph shows a curve having three pieces.

The first piece is in the interval $-1\le x<0$ . It is a line passing through the points $\left(-1,1\right)$ and $\left(0,0\right)$ . Write the equation of this line and isolate the variable $y$ from it.

  $\begin{array}{c}\frac{y-1}{x-\left(-1\right)}=\frac{0-1}{0-\left(-1\right)}\\ \frac{y-1}{x+1}=\frac{-1}{1}\\ y-1=-\left(x+1\right)\\ y=1-x-1\\ y=-x\end{array}$

The second piece is in the interval $0<x\le 1$ . It is a horizontal line at a distance of $1$ unit above the $x$ axis. Write the equation of this line as $y=1$ .

The third piece is in the interval $1<x<3$ . It is a line passing through the points $\left(1,1\right)$ and $\left(3,0\right)$ . Write the equation of this line and isolate the variable $y$ from it.

  $\begin{array}{c}\frac{y-1}{x-1}=\frac{0-1}{3-1}\\ \frac{y-1}{x-1}=\frac{-1}{2}\\ y-1=-\frac{1}{2}\left(x-1\right)\\ y=1-\frac{x}{2}+\frac{1}{2}\\ y=\frac{3}{2}-\frac{x}{2}\end{array}$

Combine the three pieces to get the piecewise defined function.

  $f\left(x\right)=\left\{\begin{array}{ll}-x,\hfill & -1\le x<0\hfill \\ 1,\hfill & 0<x\le 1\hfill \\ \frac{3}{2}-\frac{x}{2},\hfill & 1<x<3\hfill \end{array}\right.$

Conclusion:

The piecewise defined function for the given graph is $f\left(x\right)=\left\{\begin{array}{ll}-x,\hfill & -1\le x<0\hfill \\ 1,\hfill & 0<x\le 1\hfill \\ \frac{3}{2}-\frac{x}{2},\hfill & 1<x<3\hfill \end{array}\right.$ .