Question

The number of integers in the domain of $f\left(x\right)=\sqrt{\frac{\left[x\right]-2}{5-\left[x\right]}},$ where $[.]$ represents the greatest integer function, is

Answer 1

For $f$ to be defined,
$\frac{\left[x\right]-2}{5-\left[x\right]}\ge 0,\left[x\right]\ne 5$
$\Rightarrow \frac{\left[x\right]-2}{\left[x\right]-5}\le 0,x\notin [5,6)$
$\Rightarrow 2\le \left[x\right]\le 5,x\notin [5,6)$
$\Rightarrow 2\le x<6,x\notin [5,6)$
$\Rightarrow 2\le x<5$
Hence, domain of $f$ is $[2,5)$