Chapter 6.5, Problem 39AYU

In Problems 17-40, graph each function. Be sure to label key points and show at least two cycles. Use the graph to determine the domain and the range of each function.

$y=2\csc \left(\frac{1}{3}x\right)-\text{1}$

To determine

To find: Domain and range.

Answer to Problem 39AYU

Solution:

Domain is $\left\{x\text{|}x\ne 3k\pi ,k\text{ is an integer}\right\}$.

Range $\left\{y\text{|}y\le -3\text{ or }y\ge 1\right\}$.

Explanation of Solution

Given:

$y=2\csc \left(\frac{1}{3}x\right)-1$

Calculation:

The graph $y=2\csc \left(\frac{1}{3}x\right)-1$.

By the graph we determine that the domain of this function is the set of all real numbers where cosecant is defined, which excludes points $x$ where vertical asymptotes occur. Since $2\csc \left(\frac{1}{3}x\right)=\frac{2}{\sin \left(\frac{1}{3}x\right)}$ this happens when $\text{sin}\left(\frac{1}{3}x\right)=0\Rightarrow \left(\frac{1}{3}x\right)=k\pi \Rightarrow x=3k\pi$. Therefore the domain is $\left\{x\text{|}x\ne 3k\pi ,k\text{ is an integer}\right\}$. To determine the range we take note of the value of $A=2$, which will scale the underlying reciprocal sine function amplitude to 2. The given function $2\csc \left(\frac{1}{3}x\right)$ will not enter into the space of the cosine $y=2\cos \left(\frac{1}{3}x\right)$ (apart from its maxima and minima). Therefore without the vertical shift, the range is going to be $\left\{y\text{|}y\le -2\cup y\ge 2\right\}$. However the vertical shift applied this range likewise gets shifted, to $\left\{y\text{|}y\le -2-1\cup y\ge 2-1\right\}=\left\{y\text{|}y\le -3\cup y\ge 1\right\}$. This agrees with the graph provided.

Therefore the,

Domain is $\left\{x\text{|}x\ne 3k\pi ,k\text{ is an integer}\right\}$.

Range $\left\{y\text{|}y\le -3\text{ or }y\ge 1\right\}$.