Chapter 4.5, Problem 18E

Solution

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

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

Given information:

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

Explanation:

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

It can be written as $\sec x=\frac{1}{\cos x}$ .

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

The domain of $f\left(x\right)=\sec x$ is all the real number except multiples of $\frac{\pi }{2}+n\pi$ 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 increasing on $\left(0,\frac{\pi }{2}\right)\cup \left(\frac{\pi }{2},\pi \right)$ decreasing on the $\left(\pi ,\frac{3\pi }{2}\right)\cup \left(\frac{3\pi }{2},2\pi \right)$ .

Also,

  $\sec \left(-x\right)=\sec x$

So, this is an even function.

The graph is also symmetric with respect to $y$ -axis.

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 even multiples of $\pi$ and local maximum $-1$ at odd multiples of $\pi$ .

The graph has vertical asymptotes where the cosine function has zeroes. So, vertical asymptotes are the integer multiples of $\frac{\pi }{2}+n\pi$ .

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

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