Question

If $[x]$ denotes the greatest integer function, then the domain of the function $f(x)=\sqrt{\frac{x-[x]}{\log \left(x^{2}-x\right)}}$, is

• A. $(1, \infty)$
• B. $(1, \infty)-Z$
• C. $R-\left[\frac{1-\sqrt{5}}{2}, \frac{1+\sqrt{5}}{2}\right]$
• D. $\left[\frac{1-\sqrt{5}}{2}, \frac{\sqrt{5}+1}{2}\right]$

Answer 1

Answer: (C)

We have,
$f(x)=\sqrt{\frac{x-[x]}{\log \left(x^{2}-x\right)}}$
It is defined
$\log \left(x^{2}-x\right)>0 \,\,\,[\because x-[x]=[x] \geq 0, \forall x \in R]$
$\Rightarrow x^{2}-x>1 $
$\Rightarrow x^{2}-x-1>0$
$\because x^{2}-x-1=0 $
$\Rightarrow x=\frac{1 \pm \sqrt{5}}{2}$
$\therefore x \in\left(-\infty, \frac{1-\sqrt{5}}{2}\right) \cup\left(\frac{1+\sqrt{5}}{2}, \infty\right)$
$\therefore $ Domain of $f(x)=R-\left[\frac{1-\sqrt{5}}{2}, \frac{1+\sqrt{5}}{2}\right]$