The domain of the function $$f\left( x \right) = {\log _e}\left\{ {\operatorname{sgn} \left( {9 - {x^2}} \right)} \right\} + \sqrt {{{\left[ x \right]}^3} - 4\left[ x \right]} $$          (where [.] represents the greatest integer function) is :

Solution :
$$\eqalign{ & {\text{We have }}f\left( x \right) = {\log _e}\left\{ {\operatorname{sgn} \left( {9 - {x^2}} \right)} \right\} + \sqrt {{{\left[ x \right]}^3} - 4\left[ x \right]} \cr & {\text{We must have, }}\operatorname{sgn} \left( {9 - {x^2}} \right) > 0 \cr & \Rightarrow 9 - {x^2} > 0 \cr & \Rightarrow {x^2} - 9 < 0 \cr & \Rightarrow \left( {x - 3} \right)\left( {x + 3} \right) < 0 \cr & \Rightarrow - 3 < x < 3......({\text{i}}) \cr & {\text{Also }}{\left[ x \right]^3} - 4\left[ x \right] \geqslant 0 \cr & \Rightarrow \left[ x \right]\left( {{{\left[ x \right]}^2} - 4} \right) \geqslant 0 \cr & \Rightarrow \left[ x \right]\left( {\left[ x \right] - 2} \right)\left( {\left[ x \right] + 2} \right) \geqslant 0 \cr & \Rightarrow \left[ x \right] \geqslant - 2{\text{ or }}\left[ x \right]{\text{ lies between }} - 2{\text{ and}}\,0{\text{ i}}{\text{.e}}{\text{., }}\left[ x \right] = - 2,\, - 1\,{\text{or }}0 \cr & {\text{Now, }}\left[ x \right] \geqslant - 2 \Rightarrow x \geqslant 2......({\text{ii}}) \cr & \left[ x \right] = - 2 \Rightarrow - 2 \leqslant x < - 1; \cr & \left[ x \right] = - 1 \Rightarrow - 1 \leqslant x < 0; \cr & \left[ x \right] = 0 \Rightarrow 0 \leqslant x < 1 \cr & {\text{Hence, }}\left[ x \right] = - 2,\, - 1,\,0 \Rightarrow - 2 \leqslant x < 1 \cr & {\text{Hence,}}\,{D_f} = \left[ { - 2,\,1} \right) \cup \left[ {2,\,3} \right) \cr} $$