Chapter 6.5, Problem 26AYU

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.

$y=\frac{1}{2}\csc x$

To determine

To find: Domain and range.

Answer to Problem 26AYU

Solution:

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

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

Explanation of Solution

Given:

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

Calculation:

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

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 $\frac{1}{2}\csc x=\frac{1}{2\sin x}$ this happens when $\sin x=0\Rightarrow x=k\pi$. Therefore the domain is $\left\{x\text{|}x\ne k\pi ,k\text{ is an integer}\right\}$. 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 $\frac{1}{2}\csc x$ will not enter into the space of the cosine $y=\frac{1}{2}\cos x$ (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 $\left\{x\text{|}x\ne k\pi ,k\text{ is an integer}\right\}$.

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