The function $f(x) = \frac{\sec^{-1}x}{\sqrt{x - [x]}}$,where $[.]$ denotes the greatest integer less than or equal to $x$,is defined for all $x$ belonging to:

Let $f(x)=\sqrt{2-x-x^2}$ and $g(x)=\cos x$. Which of the following statements are true?
$I$. Domain of $f((g(x))^2) = \text{Domain of } f(g(x))$
$II$. Domain of $f(g(x)) + g(f(x)) = \text{Domain of } g(f(x))$
$III$. Domain of $f(g(x)) = \text{Domain of } g(f(x))$
$IV$. Domain of $g((f(x))^3) = \text{Domain of } f(g(x))$

The domain of the function $f(x) = \sqrt{2 - 2x - x^2}$ is

The domain of definition of $f(x) = \sqrt{\frac{1-|x|}{2-|x|}}$ is: (Here $(a, b) = \{x : a < x < b\}$ and $[a, b] = \{x : a \leq x \leq b\}$)

The domain of the function $f(x) = \sqrt{\cos x}$ is

The range of the function $f(x) = \log_{\sqrt{5}}(3 + \cos(\frac{3\pi}{4} + x) + \cos(\frac{\pi}{4} + x) + \cos(\frac{\pi}{4} - x) - \cos(\frac{3\pi}{4} - x))$ is: