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

The domain of the real-valued function $f(x) = \frac{\sqrt{\log_{10}\left(\frac{x}{x-2}\right)}}{\sqrt{[x]^2-5[x]+6}}$ is (where $[x]$ denotes the greatest integer function):

If $f: R \rightarrow R$ is defined by $f(x) = \frac{1}{2 - \cos 3x}$ for each $x \in R$, then the range of $f$ is

The set of all real values of $x$ for which $f(x)=\sqrt{\frac{|x|-2}{|x|-3}}$ is a well-defined function is

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

The mid-point of the domain of the function $f(x) = \sqrt{4 - \sqrt{2x + 5}}$ for real $x$ is