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