The domain of the function $f(x) = \sqrt{\log_{0.5} x!}$ is

Let the sets $A$ and $B$ denote the domain and range respectively of the function $f(x)=\frac{1}{\sqrt{\lceil x\rceil-x}}$ where $\lceil x \rceil$ denotes the smallest integer greater than or equal to $x$. Then among the statements
$(S1): A \cap B = (1, \infty) - \mathbb{N}$ and
$(S2): A \cup B = (1, \infty)$

If the function $f: R - \{ 1, - 1\} \to A$ defined by $f(x) = \frac{x^2}{1 - x^2}$ is surjective,then $A$ is equal to

The domain of the function $f(x) = \sqrt{\log_{10}\left(\frac{5x - x^2}{4}\right)}$ is:

The domain of the function $\log _{10}(x^2-5x+6)$ is

If $f: R-\{2\} \rightarrow R$ is a function defined by $f(x)=\frac{x^2-4}{x-2}$,then the range is