If ${e^x} = y + \sqrt {1 + {y^2}} $, then $y =$

Let $S=\{1,2,3,4,5,6\}$. Then the number of oneone functions $f: S \rightarrow P(S)$, where $P(S)$ denote the power set of $S$, such that $f(n) \subset f(m)$ where $n < m$ is $..................$

Range of the function , $f (x) = cot ^{-1}$ $\left( {{{\log }_{4/5}}\,\,(5\,{x^2}\,\, - \,\,8\,x\,\, + \,\,4)\,} \right)$ is :

The maximum value of function $f(x) = \int\limits_0^1 {t\,\sin \,\left( {x + \pi t} \right)} dt,\,x \in \,R$ is

The domain of definition of the function $y(x)$ given by ${2^x} + {2^y} = 2$ is

Let $f : R \rightarrow R$ be a function such that $f(x)=\frac{x^2+2 x+1}{x^2+1}$. Then