The domain of the function of $\mathrm{f}\left(\mathrm{x}\right)={\log }_{\left[\mathrm{x}\right]} \left\{\mathrm{sgn} \left({\mathrm{x}}^2\right)\right\}$ (where [.] G.I.F) is

(i) )   f(x) is defined  if$\left[x\right]⩾2\Rightarrow x\in \left[2,\infty \right]$

(ii) The function is defined if $\left(\mathrm{*}\mathrm{i}\right)\mathrm{sgn} \left(x^2\right)>0$     and (ii) $\left[\mathrm{x}\right]>0$ but $\left[x\right]\ne 1$ We know that  $\mathrm{sgn} \left(x^2\right)=\left\{\begin{array}{l}1\text{ if }{x}^2>0\\ 0\text{ if }x=0\\ -1\text{ if }{x}^2<0\end{array}\right.$

(i) Since sgn$\left(x^2\right)$ is non-negative, we have ${\mathrm{x}}^2>0\Rightarrow \mathrm{x}\in \mathrm{R}-\left\{0\right\}$

(ii) $\Rightarrow \left[\mathrm{x}\right]=2,3,4,\dots \therefore \mathrm{x}\in [2,\mathrm{\infty })$

Hence, ${\mathrm{D}}_{\mathrm{f}}=\mathrm{A}\cap \mathrm{B}=[2,\mathrm{\infty })$