Question

The domain of the function
$f\left(x\right)={\log }_{1/2}(-{\log }_2(1+\frac{1}{\sqrt[4]{4x}})-1)$ is

• A. $0<x<1$
• B. $0<x\le 1$
• C. $x\ge 1$
• D. null set

Answer 1

The correct option is A $0<x<1$

$f\left(x\right)={\log }_2(-{\log }_{1/2}(1+\frac{1}{x^{1/4}})-1)$
$itisdefinedfor$
$(-{\log }_{1/2}(1+\frac{1}{x^{1/4}})-1)>0$
$\Rightarrow ({\log }_{1/2}(1+\frac{1}{x^{1/4}})+1)<0$
$\Rightarrow \left({\log }_{1/2}(1+\frac{1}{x^{1/4}})\right)<-1$
$\Rightarrow (1+\frac{1}{x^{1/4}})>{\left(\frac{1}{2}\right)}^{-1}$
$\Rightarrow 1+\frac{1}{x^{1/4}}>2$
$\Rightarrow \frac{1}{x^{1/4}}>1$
$\Rightarrow 0<x<1$