Domain of the function $f(x) =$ $\frac{1}{{\sqrt {\ln \,{{\cot }^{ - 1}}x} }}$ is

Let $S=\{1,2,3,4\}$. Then the number of elements in the set $\{f: S \times S \rightarrow S: f$ is onto and $f(a, b)=f(b, a)$ $\geq a; \forall(a, b) \in S \times S\}$ is

The range of $f(x) = \cos (x/3)$ is

The domain of the function $f(x) = {\sin ^{ - 1}}[{\log _2}(x/2)]$ is

If $\mathrm{R}=\left\{(\mathrm{x}, \mathrm{y}): \mathrm{x}, \mathrm{y} \in \mathrm{Z}, \mathrm{x}^{2}+3 \mathrm{y}^{2} \leq 8\right\}$ is a relation on the set of integers $\mathrm{Z},$ then the domain of $\mathrm{R}^{-1}$ is 

Show that the function $f : R \rightarrow R$ given by $f ( x )= x ^{3}$ is injective.