Domain (D) and range (R) of $f\left(x\right)={\sin }^{-1}\left({\cos }^{-1}\left[x\right]\right),$ where [.] denotes the greatest integer function, is

$\mathrm{When}\left[x\right]=0, \mathrm{we} \mathrm{have} {\sin }^{-1}\left({\cos }^{-1}0\right)={\sin }^{-1}\left(\pi /2\right), \mathrm{not} \mathrm{defined}.$

$\mathrm{When}\left[x\right]=-1, \mathrm{we} \mathrm{have} {\sin }^{-1}\left({\cos }^{-1}\left(-1\right)\right)={\sin }^{-1}\left(\pi \right), \mathrm{not} \mathrm{defined}.$

$\mathrm{When}\left[x\right]=1, \mathrm{we} \mathrm{have} {\sin }^{-1}\left({\cos }^{-1}1\right)={\sin }^{-1}\left(0\right)=0.$

$\mathrm{Hence}, x\in \left[1,2\right) \mathrm{and} \mathrm{the} \mathrm{range} \mathrm{of} \mathrm{function} \mathrm{is} \left\{0\right\}.$