The domain of the function $y = \frac{1}{\sqrt{|x| - x}}$ is

The domain of the function defined by $f(x) = \frac{-5}{4x^2+1} + \sqrt{x^2-4}$ is

The domain of the function $f(x) = [\log_{10}(\frac{5x - x^2}{4})]^{1/2}$ is

Let $[x]$ denote the greatest integer not more than $x$. If $A$ and $B$ are the domains of the functions $f(x)=\frac{x-[x]}{\sqrt{|x|-x}}$ and $g(x)=\frac{x-[x]}{\sqrt{|x|+x}}$ respectively, then

If $f: R \rightarrow R$ is defined as $f(x) = \frac{x^6}{x^6+2020}$,for all $x \in R$,then the range of $f$ is .......

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 $g((f(x))^3) = \text{Domain of } f(g(x))$