The range of the function $f(x) = \frac{x}{{1 + \left| x \right|}},\,x \in R,$ is

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

Let $\mathrm{f}(\mathrm{x})$ be a polynomial of degree $3$ such that $\mathrm{f}(\mathrm{k})=-\frac{2}{\mathrm{k}}$ for $\mathrm{k}=2,3,4,5 .$ Then the value of $52-10 \mathrm{f}(10)$ is equal to :

Which of the following is correct

The period of the function $f(x) = e^{x -[x]+|cos\, \pi x|+|cos\, 2\pi x|+....+|cos\, n\pi x|}$ (where $[.]$ denotes greatest integer function); is:-

If a function $f(x)$ is such that $f\left( {x + \frac{1}{x}} \right) = {x^2} + \frac{1}{{{x^2}}};$ then  $(fof )$ $\sqrt {11} )$ =