Chapter 6.5, Problem 34AYU

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.

$y=2\cot x-\text{1}$

To determine

To find: Domain and range.

Answer to Problem 34AYU

Solution:

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

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

Explanation of Solution

Given:

$y=2\cot x-1$

Calculation:

The graph $y=2\cot x-1$.

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 $\cot x=\frac{\cos x}{\sin x}$ this happens when $\sin x=0\Rightarrow x=k\pi$. Therefore the domain is $\left\{x\text{|}x\ne k\pi ,k\text{ is an integer}\right\}$. Due to the positive and negative vertical asymptotes the range is $\left\{y\text{|}y\le -\infty ory\ge \infty \right\}$. This agrees with the graph provided. (Notice the vertical shift doesn’t change the domain or the range).

Therefore the,

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

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