Chapter 1.6, Problem 11E

(a)

To determine

To find: The period of the function $y=3\csc \left(3x+\pi \right)-2$ .

Answer to Problem 11E

The period of the function is $\frac{2\pi }{3}$ .

Explanation of Solution

Given information:

The given function is $y=3\csc \left(3x+\pi \right)-2$ .

Calculation:

The period of a function is given by $\frac{2\pi }{B}$ .

Consider the given function.

  $y=3\csc \left(3x+\pi \right)-2$

So, the period of the given function is:

  $\begin{array}{c}\frac{2\pi }{B}=3\\ B=\frac{2\pi }{3}\end{array}$

Therefore, the period of the function is $\frac{2\pi }{3}$ .

(b)

To determine

To find: The domain of the function $y=3\csc \left(3x+\pi \right)-2$ .

Answer to Problem 11E

The domain of the function is $x\ne \frac{\pi }{3}k,k\in \mathbb{Z}$ .

Explanation of Solution

Given information:

The given function is $y=3\csc \left(3x+\pi \right)-2$ .

Calculation:

Simplify the function.

  $y=\frac{3}{\sin \left(3x+\pi \right)}-2$

The function is undefined for $\sin \left(3x+\pi \right)=0$ . So,

Therefore, the domain of the function is $x\ne \frac{\pi }{3}k,k\in \mathbb{Z}$ .

(c)

To determine

To find: The range of the function $y=3\csc \left(3x+\pi \right)-2$ .

Answer to Problem 11E

The range of the function is $\left(-\infty ,-5\right)\cup \left(1,\infty \right)$ .

Explanation of Solution

Given information:

The given function is $y=3\csc \left(3x+\pi \right)-2$ .

Calculation:

Simplify the function.

  $y=\frac{3}{\sin \left(3x+\pi \right)}-2$

It is known that $-1\le \sin x\le 1$ . So, $-1\le \sin \left(3x+\pi \right)\le 1$ .

  $\text{csc}\left(3x+\pi \right)\le -1$ and $\text{csc}\left(3x+\pi \right)\ge 1$

For $\text{csc}\left(3x+\pi \right)\le -1$ ,

  $\begin{array}{c}3\csc \left(3x+\pi \right)\le -3\\ 3\csc \left(3x+\pi \right)-2\le -5\end{array}$

For $\text{csc}\left(3x+\pi \right)\ge 1$ ,

  $\begin{array}{c}3\csc \left(3x+\pi \right)\ge 3\\ 3\csc \left(3x+\pi \right)-2\ge 1\end{array}$

Therefore, the range of the function is $\left(-\infty ,-5\right)\cup \left(1,\infty \right)$ .

(d)

To determine

To graph: The function $y=3\csc \left(3x+\pi \right)-2$ .

Explanation of Solution

Given information:

The given function is $y=3\csc \left(3x+\pi \right)-2$ .

Graph:

To graph a function $y=3\csc \left(3x+\pi \right)-2$ , follow the steps using graphing calculator.

First press “ON” button on graphical calculator, press $\boxed{\text{Y}=}$ key and enter right hand side of the equation $y=3\text{cosec}\left(3x+\pi \right)-2$ after the symbol ${\text{Y}}_1$ . Enter the keystrokes $\boxed{3}\boxed{(}\boxed{1}\boxed{÷}\boxed{\text{SIN}}\boxed{3}\boxed{\text{X}}\boxed{+}\boxed{2\text{nd}}\boxed{^}\boxed{)}\boxed{)}\boxed{-}\boxed{2}$ .

The display will show the equation,

  ${\text{Y}}_{\text{1}}=3\left(1/\sin \left(3\text{X}+\pi \right)\right)-2$

Now, press the $\boxed{\text{GRAPH}}$ key and $\boxed{\text{TRACE}}$ key to produce the graph of both the equations as shown below.

  

Figure (1)

Interpretation: From the graph it can be noticed that two periods of the function are shown in the window.