Chapter 4.5, Problem 19E

Solution

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

The function $f\left(x\right)=\csc x$ has all the real number except integer multiples of $\pi$ as domain and the range is $\left(-\infty ,-1\right]\cup \left[1,\infty \right)$ , graph is continuous on the domain and the function is decreasing on $\left(0,\frac{\pi }{2}\right)\cup \left(\frac{3\pi }{2},2\pi \right)$ and increasing on the $\left(\frac{\pi }{2},\pi \right)\cup \left(\pi ,\frac{3\pi }{2}\right)$ , it is an odd function, graph is also symmetric with respect to origin, not bounded, has local minimum 1 at $\frac{\pi }{2}+2n\pi$ and local maximum $-1$ at $\frac{3\pi }{2}+2n\pi$ , vertical asymptotes are the integer multiples of has local minimum 1 at $\frac{\pi }{2}+2n\pi$ and local maximum $-1$ at $\frac{3\pi }{2}+2n\pi$ and $\underset{x\to \pm \infty }{\lim }\csc x$ does not exist.

Given information:

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

Explanation:

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

It can be written as $\csc x=\frac{1}{\sin x}$ .

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

The domain of $f\left(x\right)=\csc x$ is all the real number except integer multiples of $\pi$ and the range is $\left(-\infty ,-1\right]\cup \left[1,\infty \right)$ .

Graph the function.

  

The graph is continuous on the domain and the function is decreasing on $\left(0,\frac{\pi }{2}\right)\cup \left(\frac{3\pi }{2},2\pi \right)$ and increasing on the $\left(\frac{\pi }{2},\pi \right)\cup \left(\pi ,\frac{3\pi }{2}\right)$ .

Also,

  $\csc \left(-x\right)=-\csc x$

So, this is an odd function.

The graph is also symmetric with respect to origin.

As the range of function is $\left(-\infty ,-1\right]\cup \left[1,\infty \right)$ , the function is not bounded. The function has local minimum 1 at $\frac{\pi }{2}+2n\pi$ and local maximum $-1$ at $\frac{3\pi }{2}+2n\pi$ .

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

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

Thus, the function $f\left(x\right)=\csc x$ has all the real number except integer multiples of $\pi$ as domain and the range is $\left(-\infty ,-1\right]\cup \left[1,\infty \right)$ , graph is continuous on the domain and the function is decreasing on $\left(0,\frac{\pi }{2}\right)\cup \left(\frac{3\pi }{2},2\pi \right)$ and increasing on the $\left(\frac{\pi }{2},\pi \right)\cup \left(\pi ,\frac{3\pi }{2}\right)$ , it is an odd function, graph is also symmetric with respect to origin, not bounded, has local minimum 1 at $\frac{\pi }{2}+2n\pi$ and local maximum $-1$ at $\frac{3\pi }{2}+2n\pi$ , vertical asymptotes are the integer multiples of has local minimum 1 at $\frac{\pi }{2}+2n\pi$ and local maximum $-1$ at $\frac{3\pi }{2}+2n\pi$ and $\underset{x\to \pm \infty }{\lim }\csc x$ does not exist.