Let $f ( x )=2 x ^{ n }+\lambda, \lambda \in R , n \in N$, and $f (4)=133$, $f(5)=255$. Then the sum of all the positive integer divisors of $( f (3)- f (2))$ is

Consider the function $f: \mathbb{R} \rightarrow \mathbb{R}$ defined by

$f(x)=\frac{2 x}{\sqrt{1+9 x^2}}$. If the composition of $f, \underbrace{(f \circ f \circ f \circ \ldots \circ f)}_{10 \text { times }}(x)=\frac{2^{10} x}{\sqrt{1+9 \alpha x^2}}$, then the value of $\sqrt{3 \alpha+1}$ is equal to....................

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

For $x\,\, \in \,R\,,x\, \ne \,0,$ let ${f_0}(x) = \frac{1}{{1 - x}}$ and ${f_{n + 1}}(x) = {f_0}({f_n}(x)),$ $n\, = 0,1,2,....$  Then the value of ${f_{100}}(3) + {f_1}\left( {\frac{2}{3}} \right) + {f_2}\left( {\frac{3}{2}} \right)$ is equal to

Product of all the solution of the equation  ${x^{1 + {{\log }_{10}}x}} = 100000x$ is

Domain of the function $f(x) = {\sin ^{ - 1}}(1 + 3x + 2{x^2})$ is