Question:
Write the domain and the range of the function, $f(x)=\sqrt{x-[x]}$.
Solution:
Given, $\mathrm{f}(\mathrm{x})=\sqrt{x-[x]}$
Where [x] is the Greatest Integer Function of x.
$f(x)=\sqrt{\{x\}}$
Where {x} is fractional part of x.
The graph of $f(x)$ is
(i) dom(f)
Domain of{x} is R.
The value of the fractional part of x is always either positive or zero.
Hence domain of f(x) is R.
(ii) range(f)
Range of $\{x\}$ is $[0,1)$.
As the root value $[0,1)$ between interval lies between $[0,1)$.
Hence range of $f(x)$ is $[0,1)$.