Q42 / 160 · TS EAMCET 2023 (13 May Shift 1) · Mathematics
The domain of the real valued function
$$
f(x)=\frac{\sqrt{\log _{10}\left(\frac{x}{x-2}\right)}}{\sqrt{[x]^2-5[x]+6}} \text { is }
$$
(Here $[\mathrm{x}]$ denotes the greatest integer function)
- A$(-\infty, 0) \cup(2, \infty)$
- B$(2, \infty)$
- C$(-\infty, 2) \cup(4, \infty)$
- Correct answerD$[4, \infty)$
Solution
$f(x)=\frac{\sqrt{\log _{10} \frac{x}{x-2}}}{\sqrt{[x]^2-5[x]+6}}$ (given)
$$
\begin{aligned}
& {[x]^2-5[x]+6>0} \\
& \Rightarrow([x]-3)([x]-2)>0 \\
& \Rightarrow[x] < 2 \text { or }[x]>3 \Rightarrow x \in(-\infty, 2) \cup[4, \infty) \\
& \text { and } \log _{10} \frac{x}{x-2} \geq 0 \Rightarrow \frac{x}{x-2} \geq 1 \Rightarrow \frac{x}{x-2}-1 \\
& \Rightarrow \frac{2}{x-2} \geq 0 \quad \therefore x \in(2, \infty)
\end{aligned}
$$
Taking common solution $\therefore x \in[4, \infty)$.
Hence, the correct answer is option: 'D'.