Question

The domain of the function $f\left(x\right)=\sqrt{x^2-[x{]}^2}$ is
$($ $[.]$ represents the greatest integer function $)$

• A. $[0,\infty )$
• B. $(-\infty ,0]$
• C. $R$
• D. None of these

Answer 1

The correct option is D None of these

$f\left(x\right)=\sqrt{x^2-[x{]}^2}$
$f$ is defined if
$x^2-[x{]}^2\ge 0\cdots (1)$
$\Rightarrow (x-[x\left]\right)(x+[x\left]\right)\ge 0$
$\Rightarrow x+\left[x\right]\ge 0(\because x-[x]\ge 0\forall x\in R)$
$\Rightarrow x\ge 0$

Also, if $x\in Z,x=\left[x\right]$
So, eqn$\left(1\right)$ is satisfied for $x\in Z$ also.

Hence, the domain of $f$ is $\{x:x\ge 0\}\cup Z$