Question

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

Answer 1

$f\left(x\right)=\sqrt{x-\left[x\right]}$
$$\begin{gathered} \mathrm{Since}f\left(x\right)\mathrm{is}\mathrm{defined}\mathrm{for}\mathrm{all}\mathrm{values}\mathrm{of}\mathrm{x},x\in R. \\ \text{Or}\mathrm{dom}\left(f\right(x\left)\right)=R \\ \mathrm{Since},x-\left[x\right]=\left\{x\right\},\mathrm{which}\mathrm{is}\mathrm{the}\mathrm{fractional}\mathrm{part}\mathrm{of}\mathrm{any}\mathrm{real}\mathrm{number}x, \\ f\left(x\right)=\sqrt{\left\{x\right\}}.....\left(1\right) \\ \mathrm{We}\mathrm{know}\mathrm{that} \\ 0\le \left\{x\right\}<1 \\ \Rightarrow \sqrt{0}\le \sqrt{\left\{x\right\}}<\sqrt{1} \\ \Rightarrow 0\le f\left(x\right)<1\left\{\mathrm{from}\right(1\left)\right\} \\ \mathrm{Thus},\text{r}\mathrm{ange}\mathrm{of}f\left(x\right)\mathrm{is}[0,1). \end{gathered}$$