Chapter 4.5, Problem 20E

Solution

The function $f\left(x\right)=\tan \frac{x}{2}$ for the domain, range, continuity, increasing or decreasing behaviour, asymmetry, boundedness, extrema, asymptotes, and end behaviour.

The function $f\left(x\right)=\tan \frac{x}{2}$ has is all the real number except multiples of $n\pi$ and the range is $\left(-\infty ,\infty \right)$ , graph is continuous on the domain and the function is increasing on one period, it is an odd function, graph is also symmetric with respect to origin, not bounded, does not have local extrema, v are the multiples of $n\pi$ . It does have a horizontal asymptote. It is a periodic function, so $\underset{x\to \pm \infty }{\lim }\tan \frac{x}{2}$ do not exist.

Given information:

The function $f\left(x\right)=\tan \frac{x}{2}$ .

Explanation:

Consider the function $f\left(x\right)=\tan \frac{x}{2}$ ,

It can be written as $\tan \frac{x}{2}=\frac{\sin \frac{x}{2}}{\cos \frac{x}{2}}$ .

As the function $\cos \frac{x}{2}$ has zeroes on the integer multiple of $\pi$ , so $f\left(x\right)=\tan \frac{x}{2}$ is not defined at $n\pi$ .

The domain of $f\left(x\right)=\tan \frac{x}{2}$ is all the real number except multiples of $n\pi$ and the range is $\left(-\infty ,\infty \right)$ .

Graph the function.

  

The graph is continuous on the domain and the function is increasing on one period.

Also,

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

So, this is an odd function.

The graph is also symmetric with respect to origin.

As the range of function is $\left(-\infty ,\infty \right)$ , the function is not bounded. The function does not have local extrema.

The graph has vertical asymptotes where the $\cos \frac{x}{2}$ function has zeroes. So, vertical asymptotes are the multiples of $n\pi$ . It does have a horizontal asymptote.

As the function $f\left(x\right)=\tan \frac{x}{2}$ is a periodic function, so $\underset{x\to \pm \infty }{\lim }\tan \frac{x}{2}$ do not exist.

Thus, the function $f\left(x\right)=\tan \frac{x}{2}$ has is all the real number except multiples of $n\pi$ and the range is $\left(-\infty ,\infty \right)$ , graph is continuous on the domain and the function is increasing on one period, it is an odd function, graph is also symmetric with respect to origin, not bounded, does not have local extrema, v are the multiples of $n\pi$ . It does have a horizontal asymptote. It is a periodic function, so $\underset{x\to \pm \infty }{\lim }\tan \frac{x}{2}$ do not exist.