Chapter 6.5, Problem 30AYU

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.

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

To determine

To find: Domain and range.

Answer to Problem 30AYU

Solution:

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

Range $\left\{y\text{|}y\le -\frac{1}{2}\text{ or }y\ge \frac{1}{2}\right\}$.

Explanation of Solution

Given:

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

Calculation:

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

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

Therefore the,

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

Range $\left\{y\text{|}y\le -\frac{1}{2}\text{ or }y\ge \frac{1}{2}\right\}$.