Write the domain and range of function f(x) given by f(x)

Question:

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

Solution:

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

We know that

$[x]-x=-\{x\}$, which is the fractional part of any real number $x .$

Thus, $f(x)=\sqrt{-\{x\}}$.

Since $\{x\}$ is always a positive number, $f(x)$ is not defined for any $\mathrm{x}$.

Or $\operatorname{dom}(f)=\varphi$

Thus, range $(f)=\varphi$.