Chapter 4.6, Problem 92E

(a)

To determine

To find:The function $f\left(x\right)$ is represented by the graph.

Answer to Problem 92E

The function $f\left(x\right)$ is represented by the graph is $f\left(x\right)=\tan 2x$ .

Explanation of Solution

Given information:

The given graph is shown in figure (1).

  

Figure (1)

The given functions are as follows.

  $\begin{array}{l}\left(\text{i}\right)f\left(x\right)=\tan 2x\\ \left(\text{ii}\right)f\left(x\right)=\tan \left(x/2\right)\\ \left(\text{iii}\right)f\left(x\right)=2\tan x\\ \left(\text{iv}\right)f\left(x\right)=-\tan 2x\\ \left(\text{v}\right)f\left(x\right)=-\tan \left(x/2\right)\end{array}$

Calculation:

For $x=-\frac{\pi }{4}$ and $\frac{\pi }{4}$ , the value of the function approaches negative infinity and positive infinity respectively.

So, the vertical asymptote of the function will be at $x=-\frac{\pi }{4}$ and $\frac{\pi }{4}$ .

  $\left(\text{i}\right)f\left(x\right)=\tan 2x$

Substituting the $-\frac{\pi }{4}$ for $x$ .

  $\begin{array}{c}\tan 2x=\tan \left(2\right)\left(-\frac{\pi }{4}\right)\\ =\tan \left(-\frac{\pi }{2}\right)\\ =-\infty \end{array}$

Substituting the $\frac{\pi }{4}$ for $x$ .

  $\begin{array}{c}\tan 2x=\tan \left(2\right)\left(\frac{\pi }{4}\right)\\ =\tan \left(\frac{\pi }{2}\right)\\ =\infty \end{array}$

The value of function approaches negative infinity and positive infinity on substituting $-\frac{\pi }{4}$ and $\frac{\pi }{4}$ respectively, thus the function $f\left(x\right)=\tan 2x$ satisfy the above condition.

Similarly, check for $f\left(x\right)=2\tan x$

  $\left(\text{iii}\right)f\left(x\right)=2\tan x$

Substituting the $-\frac{\pi }{4}$ for $x$ .

  $\begin{array}{c}2\tan x=2\tan \left(-\frac{\pi }{4}\right)\\ =2\tan \left(-\frac{\pi }{4}\right)\\ =-2\end{array}$

Substituting the $\frac{\pi }{4}$ for $x$ .

  $\begin{array}{c}2\tan x=2\tan \left(\frac{\pi }{4}\right)\\ =2\tan \left(\frac{\pi }{4}\right)\\ =2\end{array}$

The value of function approaches $-2$ and $2$ on substituting $-\frac{\pi }{4}$ and $\frac{\pi }{4}$ respectively, thus the function $f\left(x\right)=2\tan x$ does not satisfy the above condition.

Therefore, the function $f\left(x\right)$ is represented by the graph is $f\left(x\right)=\tan 2x$ .

(b)

To determine

To find:The function $f\left(x\right)$ is represented by the graph.

Answer to Problem 92E

The function $f\left(x\right)$ is represented by the graph is $f\left(x\right)=\csc 4x$ .

Explanation of Solution

Given information:

The given graph is shown in figure (1).

  

Figure (1)

The given functions are as follows.

  $\begin{array}{l}\left(\text{i}\right)f\left(x\right)=\sec 4x\\ \left(\text{ii}\right)f\left(x\right)=\csc 4x\\ \left(\text{iii}\right)f\left(x\right)=\csc \left(x/4\right)\\ \left(\text{iv}\right)f\left(x\right)=\sec \left(x/4\right)\\ \left(\text{v}\right)f\left(x\right)=\csc \left(4x-\pi \right)\end{array}$

Calculation:

For $x=0$ and $\frac{\pi }{4}$ , the value of the function approaches positive infinity.

So, the vertical asymptote of the function will be at $x=0$ and $\frac{\pi }{4}$ .

Using the above graph observed that the graph is of an odd-function and $\sec x$ is an even function. Thus, we can safely rule out all the options containing sec as a function. So, that option $\left(\text{i}\right)$ and $\left(\text{iv}\right)$ can be ruled out.

  $\left(\text{ii}\right)f\left(x\right)=\csc 4x$

Substituting the $0$ for $x$ .

  $\begin{array}{c}\csc 4x=\csc \left(4\right)\left(0\right)\\ =\csc \left(0\right)\\ =\infty \end{array}$

Substituting the $\frac{\pi }{4}$ for $x$ .

  $\begin{array}{c}\csc 4x=\csc \left(4\right)\left(\frac{\pi }{4}\right)\\ =\csc \left(\pi \right)\\ =\infty \end{array}$

The value of function approaches positive infinity on substituting $x=0$ and $x=\frac{\pi }{4}$ respectively, thus the function $f\left(x\right)=\csc 4x$ satisfy the above condition.

Similarly, check for $f\left(x\right)=\csc \frac{x}{4}$

  $\left(\text{iii}\right)f\left(x\right)=\csc \frac{x}{4}$

Substituting the $0$ for $x$ .

  $\begin{array}{c}\csc \frac{x}{4}=\csc \frac{0}{4}\\ =\csc 0\\ =\infty \end{array}$

Substituting the $\frac{\pi }{4}$ for $x$ .

  $\begin{array}{c}\csc \frac{\pi }{4}=\csc \left(\frac{\frac{\pi }{4}}{4}\right)\\ =\csc \left(\frac{\pi }{16}\right)\\ \ne \infty \end{array}$

The value of function does not approaches infinityon substituting $x=0$ and $x=\frac{\pi }{4}$ respectively, thus the function $f\left(x\right)=\csc \frac{x}{4}$ does not satisfy the above condition.

Therefore, the function $f\left(x\right)$ is represented by the graph is $f\left(x\right)=\csc 4x$ .