If $f(x) = \frac{x}{{x - 1}}$, then $\frac{{f(a)}}{{f(a + 1)}} = $

The range of the function $f(x){ = ^{7 - x}}{\kern 1pt} {P_{x - 3}}$ is

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

Domain of the function $f(x) = \sqrt {2 - {{\sec }^{ - 1}}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

$( S 1): A \cap B =(1, \infty)-N$ and

$( S 2): A \cup B=(1, \infty)$

The domain of the function $f(x) = \frac{{{{\sin }^{ - 1}}(3 - x)}}{{\ln (|x|\; - 2)}}$ is