Chapter 0, Problem 39RE

To determine

To write: a piecewise formula for the given function.

Answer to Problem 39RE

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

Explanation of Solution

Given information:

The function:

  

Formula Used:

The slope of a line segment containing two points $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$ is:

  $m=\frac{y_2-y_1}{x_2-x_1}$

The equation of the line segment containing the point $\left(x_1,y_1\right)$ and with slope m is:

  $y-y_1=m\left(x-x_1\right)$

For the interval $\left[0,1\right)$ , the function is a line segment with end points $\left(1,0\right)$ and $\left(0,1\right)$ . So, the slope of this line segment is:

  $\begin{array}{c}m=\frac{y_2-y_1}{x_2-x_1}\\ =\frac{1-0}{0-1}\\ =-1\end{array}$

Then, the equation of this line segment is:

  $\begin{array}{c}y-y_1=m\left(x-x_1\right)\\ y-0=-1\left(x-1\right)\\ y=-x+1\end{array}$

And For the interval $\left[1,2\right]$ , the function is a line segment with end points $\left(1,1\right)$ and $\left(2,0\right)$ . So, the slope of this line segment is:

  $\begin{array}{c}m=\frac{y_2-y_1}{x_2-x_1}\\ =\frac{0-1}{2-1}\\ =\frac{-1}{1}\\ =-1\end{array}$

Then, the equation of this line segment is:

  $\begin{array}{c}y-y_1=m\left(x-x_1\right)\\ y-1=-1\left(x-1\right)\\ y-1=-x+1\\ y=-x+2\end{array}$

The required piecewise formula of the function is:

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