Let $f:(1,3) \rightarrow \mathrm{R}$ be a function defined by

$f(\mathrm{x})=\frac{\mathrm{x}[\mathrm{x}]}{1+\mathrm{x}^{2}},$ where $[\mathrm{x}]$ denotes the greatest

integer $\leq \mathrm{x} .$ Then the range of $f$ is

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

The function $f(x) = \;|px – q|\; + r|x|,\;x \in ( – \infty ,\;\infty )$, where $p > 0,\;q > 0,\;r > 0$ assumes its minimum value only at one point, if

Let $f : N \rightarrow R$ be a function such that $f(x+y)=2 f(x) f(y)$ for natural numbers $x$ and $y$. If $f(1)=2$, then the value of $\alpha$ for which

$\sum \limits_{k=1}^{10} f(\alpha+k)=\frac{512}{3}\left(2^{20}-1\right)$ holds, is

Let $\mathrm{f}: \mathrm{R} \rightarrow \mathrm{R}$ be a function which satisfies $\mathrm{f}(\mathrm{x}+\mathrm{y})=\mathrm{f}(\mathrm{x})+\mathrm{f}(\mathrm{y}) \forall \mathrm{x}, \mathrm{y} \in \mathrm{R} .$ If $\mathrm{f}(1)=2$ and $g(n)=\sum \limits_{k=1}^{(n-1)} f(k), n \in N$ then the value of $n,$ for which $\mathrm{g}(\mathrm{n})=20,$ is