The domain of ${\sin ^{ – 1}}({\log _3}x)$ is

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

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

Range of $f(x) = sin^{-1} (\sqrt {x^2 + x +1})$ is –

Let $f :R \to R$ be defined by $f(x)\,\, = \,\,\frac{x}{{1 + {x^2}}},\,x\, \in \,R.$ Then the range of $f$ is