The domain of the function $f(x) = \sin^{-1}\left(\frac{x^{2}-3x+2}{x^{2}+2x+7}\right)$ is:

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

The domain of $f(x) = \frac{1}{{\sqrt {{{\log }_{\frac{\pi }{4}}}({{\sin }^{ - 1}}x) - 1} }}$ is:

Let $f$ be a real valued function defined by $f(x) = \sin^{-1} \left( \frac{1 - |x|}{3} \right) + \cos^{-1} \left( \frac{|x| - 3}{5} \right)$. Then the domain of $f(x)$ is given by:

Let $[\cdot]$ denote the greatest integer function. If the domain of the function $f(x) = \cos^{-1} \left( \frac{4x+2[x]}{3} \right)$ is $[\alpha, \beta]$, then $12(\alpha + \beta)$ is equal to:

If $y = \operatorname{cosec}^{-1}(x)$ and $\frac{dy}{dx} = \frac{-1}{|x| \sqrt{x^2-1}}$,then