Chapter 6.5, Problem 22AYU

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.

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

To determine

To find: Domain and range.

Answer to Problem 22AYU

Solution:

Domain: $\left\{x\text{|}x\ne \left(2k+1\right)\pi ,k\text{ is an odd integer}\right\}$.

Range: $\left(-\infty ,\infty \right)$.

Explanation of Solution

Given:

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

Calculation:

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

By the graph we determine that the domain of this function is the set of all real numbers where tangent is defined, which excludes points $x$ where vertical asymptotes occur.

Since $\tan \left(\frac{1}{2}x\right)=\frac{\text{sin}\left(\frac{1}{2}x\right)}{\text{cos}\left(\frac{1}{2}x\right)}$ this happens when $\text{cos}\left(\frac{1}{2}x\right)=0\Rightarrow \left(\frac{1}{2}x\right)=\frac{\left(2k+1\right)\pi }{2}\Rightarrow$.

$x=\left(2k+1\right)\pi$

Therefore the,

Domain: $\left\{x\text{|}x\ne \left(2k+1\right)\pi ,k\text{ is an odd integer}\right\}$.

Range: $\left(-\infty ,\infty \right)$.