If $f: R \rightarrow R$ and $g: R \rightarrow R$ are given by $f(x)=|x|$ and $g(x)=[x]$ for each $x \in R$, then $\{x \in R: g(f(x)) \leq f(g(x))\}$ is equal to

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

Which of the four statements given below is different from the others?

Let $f : R \rightarrow R$ and $g : R \rightarrow R$ be two functions defined by $f(x)=\log _{e}(x^{2}+1)-e^{-x}+1$ and $g(x)=\frac{1-2e^{2x}}{e^{x}}$. Then,for which of the following range of $\alpha$,the inequality $f(g(\frac{(\alpha-1)^{2}}{3})) > f(g(\alpha-\frac{5}{3}))$ holds?

The function $f(x) = [|x|] - |[x]|$ where $[x]$ denotes the greatest integer function:

The probability that a relation $R$ from $\{x, y\}$ to $\{x, y\}$ is both symmetric and transitive is equal to: