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 $f(x)=\frac{x-1}{x+1}, x \in R-\{0,-1,1)$. If $f^{a+1}(x)=f\left(f^{n}(x)\right)$ for all $n \in N$, then $f^{\prime}(6)+f(7)$ is equal to

Let $f (x) = a^x (a > 0)$ be written as $f( x) = f_1( x) + f_2( x)$ , where $f_1( x)$ is an even function and $f_2( x)$ is an odd function. Then $f_1( x + y) + f_1( x - y )$ equals

The range of values of the function $f\left( x \right) = \frac{1}{{2 - 3\sin x}}$ is

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

Domain of the function $f(x)\,=\,\frac{1}{{\sqrt {(x + 1)({e^x} - 1)(x - 4)(x + 5)(x - 6)} }}$