B Advice to obtain the domain of compound functions

[mcastillo356]

TL;DR Summary

I'm familiar to this ground, but the function composition I introduce is difficult for me

Hi PF

I have a quote from Spanish 6th edition of "Calculus", by Robert A. Adams, and some queries. I translate it this way:"The inverse of secondary trigonometric functions can easily be calculated with the reciprocal function. For example
DEFINITION 13 The inverse function of secant $sec^{-1} x$ (or $\mbox{arcsec}x$)
$$sec^{-1}=cos^{-1}\left({\dfrac{1}{x}}\right)\quad for\;|x|\geq 1$$
The domain of $\sec^{-1}$ is the union of intervals $(-\infty,-1]\cup{[1,\infty)}$ and $[0,\dfrac{\pi}{2})\cup{(\dfrac{\pi}{2},\pi)}$. The graph of $y=sec^{-1}x$ is shown in Figure 3.25(b)(*). Is the reflection respect to the line $y=x$ of the part of $\sec x$ for $x$ between 0 and $\pi$. Additionally
$$\sec(\sec^{-1}x)=\sec\left({\cos^{-1}\left({\dfrac{1}{x}}\right)}\right)
=\dfrac{1}{\cos\left({\cos^{-1}\left({\dfrac{1}{x}}\right)}\right)}=\dfrac{1}{\dfrac{1}{x}}=x\qquad{\mbox{for}\;|x|\geq 1}$$Up to now I've got to deal only with very easy compound functions. This quote represents a qualitative step forward. The domains and ranges are shown, but I would like to know: what if I had to do it by myself, if I was given only the identities, and had to manage to describe the domain and range of, suppose, the one at DEFINITION 13?

$$cos^{-1}\left({\dfrac{1}{x}}\right)$$

With no other help but the knowledge of the domain of $y=\mbox{arcos}(x)$, $(-1\leq x\leq 1)$, and $\mathbb{R}\setminus{\{0\}}$ for $\dfrac{1}{x}$

As well, isn't there a mistake, a typo, at the sentence "Is the reflection respect to the line $y=x$ of the part of $\sec x$ for $x$ between 0 and $\pi$"? Shouldn't be "of the part of $sec^{-1} x$"?.

(*)Attached image

Attempt: Pure speculation; don't know why, but I've come across this statement: domain shouldn't be the intersection of the domain of $y=\cos x$ and the domain of the inverse function of $y=\dfrac{1}{x}$?

 

[mcastillo356]

I've typed the formulas with #### and $$$$. Why didn't I post successfully?.

 

[fresh_42]

I've typed the formulas with #### and $$$$. Why didn't I post successfully?.

You have simple forgotten a single "#" somewhere early. That was all. I corrected it.