Chapter 6.5, Problem 33AYU

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.

$y=\tan \left(\frac{1}{4}x\right)\text{+}\text{1}$

To determine

To find: Domain and range.

Answer to Problem 33AYU

Solution:

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

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

Explanation of Solution

Given:

$y=\text{ tan}\left(\frac{1}{4}x\right)+1$

Calculation:

The graph $y=\text{ tan}\left(\frac{1}{4}x\right)+1$.

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 $\tan \left(\frac{1}{4}x\right)=\frac{\text{sin}\left(\frac{1}{4}x\right)}{\cos \left(\frac{1}{4}x\right)}$ this happens when $\cos \left(\frac{1}{4}x\right)=0\Rightarrow \left(\frac{1}{4}x\right)=\frac{\left(2k+1\right)\pi }{2}\Rightarrow x=2\left(2k+1\right)\pi$. Therefore the domain is $\left\{x\text{|}x\ne 2\left(2k+1\right)\pi , k\text{is}\text{an}\text{integer}\right\}$}. Due to the positive and negative vertical asymptotes the range is $\left\{y\text{|}y\le -\infty \text{or}y\ge \infty \right\}$. This agrees with the graph provided. (Notice the vertical shift doesn’t change the domain or the range).

Therefore the,

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

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