If $P(S)$ denotes the set of all subsets of a given set $S, $ then the number of one-to-one functions from the set $S = \{ 1, 2, 3\}$ to the set $P(S)$ is

Let $f(x) = sin\,x,\,\,g(x) = x.$

Statement $1:$ $f(x)\, \le \,g\,(x)$ for $x$ in $(0,\infty )$

Statement $2:$ $f(x)\, \le \,1$ for $(x)$ in $(0,\infty )$ but $g(x)\,\to \infty$ as $x\,\to \infty$

Show that the function $f: R \rightarrow R$ defined as $f(x)=x^{2},$ is neither one-one nor onto.

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

The domain of $f(x) = \frac{1}{{\sqrt {{{\log }_{\frac{\pi }{4}}}({{\sin }^{ - 1}}x) - 1} }}$,is

Let ${a_2},{a_3} \in R$ such that $\left| {{a_2} - {a_3}} \right| = 6$ and $f\left( x \right) = \left| {\begin{array}{*{20}{c}}
1&{{a_3}}&{{a_2}}\\
1&{{a_3}}&{2{a_2} - x}\\
1&{2{a_3} - x}&{{a_2}}
\end{array}} \right|,x \in R.$ Then the greatest value of $f(x)$ is