Question

The domain of $; f(x)=\sqrt{\cos (\sin x)}+\sqrt{\log _{x}\{x\}} ;\{\cdot\}$ denote the fractional part, is

• A. $[1, \pi)$
• B. $(0,2 \pi)-[1, \pi)$
• C. $\left(0, \frac{\pi}{2}\right)-\{1\}$
• D. $(0,1)$

Answer 1

Answer: (D)

$f(x)=\sqrt{\cos (\sin x)}+\sqrt{\log _{x}\{x\}}$
Domain $\cos (\sin x) \geq 0\{x\}>0, x>0, x \neq 1, \log _{x}\{x\} \geq 0$
(i) $ \cos (\sin x) \geq 0$ for all $x, x \in R [-1,1]$
(ii) $\{x\}>0, x \notin$ Int.
(iii) $x>0, x \in(0, \infty)$
(iv) $x \neq 1$
(v) $\log _{x}\{x\} \geq 0 \Rightarrow 1>f(x) \geq 0$
so $1>x \geq 0 \log _{x} f(x)$ is positive $x \in[0,1) \Rightarrow x \in(0,1)$