$ \sec(\sec^{-1}(x)) $ when $x$ belongs to $(-1,0)$

Question

$ \sec(\sec^{-1}(x))=x $ when $x$ lies in $ (-\infty,-1] \cup [1,\infty). $

But when $x$ lies in $(-1,0),$ what will this function return?

My textbook says it becomes $-x$, but I don't understand how.

Answer 1

You are right

$\sec(\sec^{-1}(x))=x$ when $x\in(-\infty,1]\bigcup[1,\infty)$

This is the domain of the function, where the points are defined. Any points where $x$ lies between $(-1,1)$ is undefined.