Question

Find the domain of the following function:

$f\left(x\right)=\frac{1}{\sqrt{{\left[x\right]}^2-\left[x\right]-6}}$,where $[\cdot ]$ denotes the greatest integer function.

• A. $(-\infty ,-2)\cup [4,\infty )$
• B. $(\infty ,-2]\cup [4,\infty )$
• C. $(-\infty ,-2)\cup (4,\infty )$
• D. $(-\infty ,2)\cup [4,\infty )$

Answer 1

The correct option is A $(-\infty ,-2)\cup [4,\infty )$

$f\left(x\right)$ is defined for
${\left[x\right]}^2-\left[x\right]-6>0\Rightarrow (\left[x\right]-3)(\left[x\right]+2)>0$

$\Rightarrow \left[x\right]<-2$ or $\left[x\right]>3$

But $\left[x\right]<-2\Rightarrow \left[x\right]=-3,-4,-\mathrm{5...}$
$\therefore$ $x<-2$

Also, $\left[x\right]>3\Rightarrow \left[x\right]=4,5,\mathrm{6...}$

$\therefore x\ge 4$

Hence, Domain of $f\left(x\right)=(-\infty ,-2)\cup [4,\infty )$

$\therefore Ans.A$