Question

The domain of the function, $f\left(x\right)=\frac{\sqrt{\sin x}}{\sqrt{(x-2)(8-x)}}$ is

• A. $[0,\pi ]\cup [2\pi ,8)$
• B. $(2,\pi ]\cup [2\pi ,8)$
• C. $(2,8)$
• D. $(0,8)$

Answer 1

The correct option is B $(2,\pi ]\cup [2\pi ,8)$

Given the function $f\left(x\right)=\frac{\sqrt{\sin x}}{\sqrt{(x-2)(8-x)}}$.

Now $f\left(x\right)$ is defined for $\sin x\ge 0$.......(1) and $(x-2)(8-x)>0$ or, $(x-2)(x-8)<0$.......(2).

From (2) we get,

$x\in (2,8)$......(3)

Form (1) we get,

$x\in [0,\pi ],[2\pi ,3\pi ],....$.......(4).

Combining (3) and (4) we get,

$x\in (2,\pi ]\cup [2\pi ,8)$.