Question

The domain of $f\left(x\right)=\sqrt{\cos \left(\sin x\right)}+\sqrt{{\log }_x\left\{x\right\}}$ (where $\{.\}$ is fractional part of $x$) is

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

Answer 1

The correct option is D $(0,1)$

$f\left(x\right)=\sqrt{cos\left(sinx\right)}+\sqrt{lo{g}_x\left\{x\right\}}$

$-1⩽sinx⩽1\forall xϵR$

$\Rightarrow 0<cos\left(sinx\right)⩽1\forall xϵR$

For function to be real

$lo{g}_x\left\{x\right\}>0$ and $\left\{x\right\}\ne 0$ and $x>0$

$\Rightarrow xϵ(0,1)$

$\therefore$ Range of f is (0,1)