The domain of definition of the function $f(x) = \sqrt{1 + \log_{e}(1 - x)}$ is

$A$ and $B$ are subsets of $R$. Every element $x$ of $A$ is mapped to an element of $B$ by the rule,$y(x) = \begin{cases} \frac{5x}{(x-3)(x+3)} & \text{if } x \neq -1 \\ -1 & \text{if } x = -1 \end{cases}$,then $A =$

Let $A = \{9, 10, 11, 12, 13\}$ and let $f: A \rightarrow N$ be defined by $f(n) = \text{the highest prime factor of } n$. Find the range of $f$.

For $f(x) = \frac{\sin \pi[x]}{1+[x]} + \frac{x}{2+3x}$,where $[x]$ denotes the greatest integer function,the domain and range in $R$ are respectively

Let $f : R \rightarrow R$ be a function defined by $f(x) = \log_{\sqrt{m}}\{\sqrt{2}(\sin x - \cos x) + m - 2\}$,for some $m$,such that the range of $f$ is $[0, 2]$. Then the value of $m$ is $............$

The range of the function $f(x) = \frac{x-3}{5-x}, x \neq 5$ is