If $f(x) = \log_{e}\left(\frac{1-x}{1+x}\right)$,$|x| < 1$,then $f\left(\frac{2x}{1+x^2}\right)$ is equal to

Let $R = \{(1, 3), (2, 2), (3, 2)\}$ and $S = \{(2, 1), (3, 2), (2, 3)\}$ be two relations on set $A = \{1, 2, 3\}$. Then $RoS =$

For equality of functions $f$ and $g$,which of the following conditions must be satisfied?
$(i)$ $\text{domain of } f = \text{domain of } g$
(ii) $f(x) = g(x)$ for all $x$ in the domain
(iii) $x \in \text{domain of } f$

If $a+\alpha=1, b+\beta=2$ and $af(x)+\alpha f\left(\frac{1}{x}\right)=bx+\frac{\beta}{x}$ for $x \neq 0$,then the value of the expression $\frac{f(x)+f\left(\frac{1}{x}\right)}{x+\frac{1}{x}}$ is ..... .

Let $[x]$ denote the integral part of $x \in R$. Let $g(x) = x - [x]$. Let $f(x)$ be any continuous function with $f(0) = f(1)$. Then the function $h(x) = f(g(x))$:

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