Question

The exhaustive domain of \(f(x)=\cot^{-1}\left(\dfrac{x}{\sqrt{x^2-\left[x^2 \right ]}} \right )\) is
(where [.] denotes the greatest integer function)

Answer 1

\(f(x)=\cot^{-1}\left(\dfrac{x}{\sqrt{x^2-\left[x^2 \right ]}} \right )\)

for \(f(x)\) to be defined, \(x^2-\left[x^2 \right ]>0\)
\(\therefore x^2\) should not be an integer
$x^2 \not= n,~n\in\mathbb{W} $
$\Rightarrow x\not=\pm \sqrt{n},~n\in\mathbb{W}$
Hence, $x\in \mathbb{R}-\{x:x=\pm\sqrt n,~n\in\mathbb{W}\}$