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