Chapter 0.2, Problem 42E

To determine

To Write:

A piecewise-defined function for the given graph.

Answer to Problem 42E

  $f\left(x\right)=\left\{\begin{array}{ll}2,\hfill & 0\le x<1\hfill \\ 0,\hfill & 1\le x<2\hfill \\ 2\hfill & 2\le x<3\hfill \\ 0\hfill & 3\le x\le 4\hfill \end{array}\right.$ .

Explanation of Solution

Given:

The graph:

  

Concepts Used:

Reading the graph of a piecewise function.

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

Calculations:

The given graph shows a curve having four pieces.

The first piece is in the interval $0\le x<1$ . It is a horizontal line at a distance of $2$ units above the $x$ axis. Write the equation of this line as $y=2$ .

The second piece is in the interval $1\le x<2$ . It is a horizontal line at a distance of $0$ units above the $x$ axis. Write the equation of this line as $y=0$ .

The third piece is in the interval $2\le x<3$ . It is a horizontal line at a distance of $2$ units above the $x$ axis. Write the equation of this line as $y=2$ .

The second piece is in the interval $3\le x\le 4$ . It is a horizontal line at a distance of $0$ units above the $x$ axis. Write the equation of this line as $y=0$ .

Combine the four pieces to get the piecewise defined function.

  $f\left(x\right)=\left\{\begin{array}{ll}2,\hfill & 0\le x<1\hfill \\ 0,\hfill & 1\le x<2\hfill \\ 2\hfill & 2\le x<3\hfill \\ 0\hfill & 3\le x\le 4\hfill \end{array}\right.$

Conclusion:

The piecewise defined function for the given graph is $f\left(x\right)=\left\{\begin{array}{ll}2,\hfill & 0\le x<1\hfill \\ 0,\hfill & 1\le x<2\hfill \\ 2\hfill & 2\le x<3\hfill \\ 0\hfill & 3\le x\le 4\hfill \end{array}\right.$ .