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

Which of the following is function

If $f(x) = \log \left[ {\frac{{1 + x}}{{1 – x}}} \right]$, then $f\left[ {\frac{{2x}}{{1 + {x^2}}}} \right]$ is equal to

Let $N$ be the set of positive integers. For all $n \in N$, let $f_n=(n+1)^{1 / 3}-n^{1 / 3} \text { and }$ $A=\left\{n \in N: f_{n+1}<\frac{1}{3(n+1)^{2 / 3}} < f_n\right\}$ Then,

If $f(x)$ satisfies the relation $f\left( {\frac{{5x – 3y}}{2}} \right) = \frac{{5f(x) – 3f(y)}}{2}\forall x,y\, \in \,R$ and $f(0)=1, f'(0)=2$ then the period of $sin(f(x))$ is