Define a function $f(x)=\frac{16 x^2-96 x+153}{x-3}$ for all real $x \neq 3$. The least positive value of $f(x)$ is

Domain of the function $f(x) = \frac{{{x^2} – 3x + 2}}{{{x^2} + x – 6}}$ 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

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

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

Let $f(x) = (1 + {b^2}){x^2} + 2bx + 1$ and $m(b)$ the minimum value of $f(x)$ for a given $b$. As $b$ varies, the range of $m(b)$ is