Chapter 6.5, Problem 36AYU

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.

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

To determine

To find: Domain and range.

Answer to Problem 36AYU

Solution:

Domain is $\left\{x\text{|}x\ne \frac{2}{3}k,k\text{ is an integer}\right\}$.

Range $=\left\{y│y\le -1\cup y\ge 1\right\}$.

Explanation of Solution

Given:

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

Calculation:

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

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 $\csc \left(\frac{3\pi }{2}x\right)=\frac{1}{\sin \left(\frac{3\pi }{2}x\right)}$ this happens when $\sin \left(\frac{3\pi }{2}x\right)=0\Rightarrow \left(\frac{3\pi }{2}x\right)=k\pi \Rightarrow x=\frac{2}{3}k$. Therefore the domain is $\left\{x\text{|}x\ne \frac{2}{3}k,k\text{ is an integer}\right\}$. To determine the range we take note of the value of $A=1$, which will scale the underlying reciprocal sine function amplitude to 1. The given function $\csc \left(\frac{3\pi }{2}x\right)$ will not enter into the space of the cosine $y=\cos \left(\frac{3\pi }{2}x\right)$ (apart from its maxima and minima). Therefore the range $\left\{y│y\le -1\cup y\ge 1\right\}$. This agrees with the graph provided.

Therefore the,

Domain is $\left\{x\text{|}x\ne \frac{2}{3}k,k\text{ is an integer}\right\}$.

Range $=\left\{y│y\le -1\cup y\ge 1\right\}$.