Q42 / 160 · TS EAMCET 2019 (03 May Shift 1) · Mathematics
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$(1, \infty)-Z$
- Correct answerC$R-\left[\frac{1-\sqrt{5}}{2}, \frac{1+\sqrt{5}}{2}\right]$
- D$\left[\frac{1-\sqrt{5}}{2}, \frac{\sqrt{5}+1}{2}\right]$
Solution
We have,
$$
f(x)=\sqrt{\frac{x-[x]}{\log \left(x^2-x\right)}}
$$
It is defined
$$
\log \left(x^2-x\right)>0 \quad[\because x-[x]=[x] \geq 0, \forall x \in R]
$$
$$
\begin{aligned}
& \Rightarrow x^2-x>1 \Rightarrow x^2-x-1>0 \\
& \because \quad x^2-x-1=0 \Rightarrow x=\frac{1 \pm \sqrt{5}}{2} \\
& \therefore \quad \quad x \in\left(-\infty, \frac{1-\sqrt{5}}{2}\right) \cup\left(\frac{1+\sqrt{5}}{2}, \infty\right) \\
& \therefore \text { Domain of } f(x)=R-\left[\frac{1-\sqrt{5}}{2}, \frac{1+\sqrt{5}}{2}\right]
\end{aligned}
$$
Hence, the correct answer is option: 'C'.