Domain of $\sin^{-1}$ is . . . . . . .

If $A = \{x \in R : \sin^{-1}(\sqrt{x^2+x+1}) \in [-\frac{\pi}{2}, \frac{\pi}{2}]\}$ and $B = \{y \in R : y = \sin^{-1}(\sqrt{x^2+x+1}), x \in A\}$,then which of the following is true?

The domain of $f(x) = \sin^{-1}\left[\log_{2}\left(\frac{x}{2}\right)\right]$ is

The domain of the function $f(x) = \sin^{-1}\left(\frac{|x|+5}{x^2+1}\right)$ is $(-\infty, -a] \cup [a, \infty)$. Then $a$ is equal to

The domain of the function $f(x) = \sin^{-1}[\log_2(x/2)]$ is

The range of $\sec ^{-1} x$ is