Chapter 1, Problem 33RE

(a)

To determine

To find: The domain of the function $y=2\sin \left(3x+\pi \right)-1$ .

Answer to Problem 33RE

The domain of the function $y=2\sin \left(3x+\pi \right)-1$ is the set of real numbers or the interval $\left(-\infty ,\infty \right)$ .

Explanation of Solution

Given information:

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

Calculation:

The function $f\left(x\right)=\sin x$ is defined for all the real values of $x$ .

So, the function $y=2\sin \left(3x+\pi \right)-1$ is defined for all the real values of $x$ .

Therefore, the domain of the function $y=2\sin \left(3x+\pi \right)-1$ is the set of real numbers or the interval $\left(-\infty ,\infty \right)$ .

(b)

To determine

To find: The range of the function $y=2\sin \left(3x+\pi \right)-1$ .

Answer to Problem 33RE

The range of the function $y=2\sin \left(3x+\pi \right)-1$ is $\left[-3,1\right]$ .

Explanation of Solution

Given information:

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

Calculation:

The function $y=\sin \left(3x+\pi \right)$ is greater than or equal to $-1$ and less than or equal to $1$ .

Multiply all sides by $2$ and then subtract $1$ from all the sides.

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

Therefore, the range of the function $y=2\sin \left(3x+\pi \right)-1$ is the interval $\left[-3,1\right]$ .

(c)

To determine

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

Explanation of Solution

Given information:

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

Graph:

To graph a function $y=2\sin \left(3x+\pi \right)-1$ , 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 $y=2\sin \left(3x+\pi \right)-1$ after the symbol ${\text{Y}}_1$ . Enter the keystrokes $\boxed{2}\boxed{\text{SIN}}\boxed{3}\boxed{\text{X}}\boxed{+}\boxed{2\text{nd}}\boxed{^}\boxed{)}\boxed{-}\boxed{1}$ .

The display will show the equation,

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

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 above graph it can be observed that the graph the function $y=2\sin \left(3x+\pi \right)-1$ is shifted downwardscomparing to its parent function.