Chapter 4.6, Problem 78E

(a)

To determine

To find: The function that represents the graph.

Answer to Problem 78E

The required period is $\tan 2x$ and (a) matches to (i).

Explanation of Solution

Given:

The given graph is shown in Figure 1

  

Figure 1

Calculation:

From the above figure the asymptotes occurs at $x=\frac{\pi }{4}$ and $x=\frac{-\pi }{4}$ with the period of $\frac{\pi }{2}$ .

The period of $\tan kx=\frac{\pi }{x}$ , so the function of the period $\tan \left(2x\right)=\frac{\pi }{2}$ that is in the positive direction. Thus, the remaining function is not in the given period.

Thus, the required period is $\tan 2x$ and (a) matches to (i).

(b)

To determine

To find: The function that represents the graph.

Answer to Problem 78E

The required period is $\csc 4x$ and (b) matches to (ii).

Explanation of Solution

Given:

The given graph is shown in Figure 1

  

Figure 2

Calculation:

From the above figure the asymptotes occurs at $x=\frac{\pi }{4}$ and $x=\frac{-\pi }{4}$ with the period of $\frac{\pi }{2}$ .

The period of $cs\left(bx-c\right)$ , so the function of the period is $\frac{b}{2\pi }$ .

Thus, the remaining functions $\csc \left(\frac{x}{4}\right),\sec \left(\frac{x}{4}\right)$ is not in the given period.

The function that is represented by the graph is $\csc 4x$ .

Thus, the required period is $\csc 4x$ and (b) matches to (ii).