Question

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

• A. $R-\{x:x=\pm n,n\in N\}$
• B. $R-\{x:x=\pm n,n\in W\}$
• C. $R-\{x:x=\pm \sqrt{n},n\in N\}$
• D. $R-\{x:x=\pm \sqrt{n},n\in W\}$

Answer 1

The correct option is D $R-\{x:x=\pm \sqrt{n},n\in W\}$

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

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