Chapter 0, Problem 40RE

(a)

To determine

To calculate : The piecewise formula for the function.

Answer to Problem 40RE

The piecewise formula for the given function is $y=f\left(x\right)=\left\{\begin{array}{l}\frac{5}{2}x,0\le x<2\\ 10-\frac{5}{2}x,2\le x\le 4\end{array}\right\}$ .

Explanation of Solution

Given information:

The graph

  

Formula Used:

Slope of line $\frac{y_2-y_1}{x_2-x_1}$

The point slope form of the equation of line is $y-y_1=m\left(x-x_1\right)$ .

Calculation:

For the given graph we can observe the following points:

There is a line through $\left(0,0\right)\text{ and }\left(2,5\right)$ .

Therefore the slope of the line can be found to be,

  $m=\frac{5-0}{2-0}=\frac{5}{2}$

So, using the point slope form of the equation $y-y_1=m\left(x-x_1\right)$ , we obtain

  $\begin{array}{l}y-0=\frac{5}{2}\left(x-0\right)\\ \Rightarrow y=\frac{5}{2}x\end{array}$

The equation of this line would be $y=\frac{5}{2}x$ .

Whereas the other line passes through $\left(2,5\right)\text{ and }\left(4,0\right)$ .

Therefore the slope of the line can be found to be,

  $\begin{array}{l}m=\frac{0-5}{4-2}\\ \text{ =}\frac{-5}{2}\\ \text{ =}\frac{5}{2}\end{array}$

So, using the point slope form of the equation $y-y_1=m\left(x-x_1\right)$ , we obtain

  $\begin{array}{l}y-5=\frac{5}{2}\left(x-2\right)\\ \Rightarrow y=-\frac{5}{2}\left(x-2\right)+5\\ \Rightarrow y=-\frac{5}{2}x+10\\ \Rightarrow y=10-\frac{5}{2}x\end{array}$

The equation of this line would be $y=10-\frac{5}{2}x$ .

Now combining both the equations according to their conditions appended to them, we get the following function:

  $y=f\left(x\right)=\left\{\begin{array}{l}\frac{5}{2}x,0\le x<2\\ 10-\frac{5}{2}x,2\le x\le 4\end{array}\right\}$

Therefore, the piecewise formula for the function is

  $y=f\left(x\right)=\left\{\begin{array}{l}\frac{5}{2}x,0\le x<2\\ 10-\frac{5}{2}x,2\le x\le 4\end{array}\right\}$ .

(b)

To determine

To calculate : Range of the given function.

Answer to Problem 40RE

The range of the given function is

Explanation of Solution

Given information:

The function $y=2\sin \left(3x+\pi \right)-1$

Calculation:

(c)

To determine

To graph : The given function.

Explanation of Solution

Given information:

The function $y=2\sin \left(3x+\pi \right)-1$

Graph:

  

Interpretation:

The domain of the given function is:

The range of the given function is: