Question

Write the domain and range of function f(x) given by $f\left(x\right)=\sqrt{\left[x\right]-x}$.

Answer 1

$f\left(x\right)=\sqrt{\left[x\right]-x}$

$$\begin{gathered} \mathrm{We}\mathrm{know}\mathrm{that} \\ \left[x\right]-x=-\left\{x\right\},\mathrm{which}\mathrm{is}\mathrm{the}\mathrm{fractional}\mathrm{part}\mathrm{of}\mathrm{any}\mathrm{real}\mathrm{number}x. \\ \mathrm{Thus},f\left(x\right)=\sqrt{-\left\{x\right\}}. \\ \mathrm{Since}\left\{x\right\}\mathrm{is}\mathrm{always}\mathrm{a}\mathrm{positive}\mathrm{number},f\left(x\right)\mathrm{is}\mathrm{not}\mathrm{defined}\mathrm{for}\mathrm{any}\mathrm{x}. \\ \text{Or}\mathrm{dom}\left(f\right)=\phi \\ \mathrm{Thus},\mathrm{range}\left(f\right)=\phi . \end{gathered}$$