Chapter 4.5, Problem 17E

Solution

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

The function $f\left(x\right)=\cot x$ has all the real number except multiples of $\pi$ as domain, the range is $\left(-\infty ,\infty \right)$ , graph is continuous on the domain and the function is decreasing on the domain, it is an odd function, symmetric with respect to origin, not bounded, does not have any local extrema, vertical asymptotes are the integer multiples of $\pi$ and $\underset{x\to \pm \infty }{\lim }\cot x$ does not exist.

Given information:

The function $f\left(x\right)=\cot x$ .

Explanation:

Consider the function $f\left(x\right)=\cot x$ ,

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

As the function $\sin x$ has zeroes on the integer multiple of $\pi$ , so $f\left(x\right)=\cot x$ is not defined at $n\pi$ .

The domain of $f\left(x\right)=\cot x$ is all the real number except multiples of $\pi$ and the range is $\left(-\infty ,\infty \right)$ .

Graph the function.

  

The graph is continuous on the domain and the function is decreasing on the domain.

Also,

  $\cot \left(-x\right)=-\cot x$

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 any local extrema.

The graph has vertical asymptotes where the sine function has zeroes. So, vertical asymptotes are the integer multiples of $\pi$ .

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

Thus, the function $f\left(x\right)=\cot x$ has all the real number except multiples of $\pi$ as domain, the range is $\left(-\infty ,\infty \right)$ , graph is continuous on the domain and the function is decreasing on the domain, it is an odd function, symmetric with respect to origin, not bounded, does not have any local extrema, vertical asymptotes are the integer multiples of $\pi$ and $\underset{x\to \pm \infty }{\lim }\cot x$ does not exist.