Question

If domain of the function $f\left(x\right)={\text{log}}_2(-{\text{log}}_{\frac{1}{2}}(1+\frac{1}{\sqrt[4]{4x}})-1)$ is $(a,b)$ then $a+b$ =

Answer 1

We have $f\left(x\right)={\text{log}}_2(-{\text{log}}_{\frac{1}{2}}(1+\frac{1}{\sqrt[4]{4x}})-1)$
$f\left(x\right)$ is defined if
$-{\text{log}}_{\frac{1}{2}}(1+\frac{1}{\sqrt[4]{4x}})-1>0,x>0$
$\Rightarrow {\text{log}}_{\frac{1}{2}}(1+\frac{1}{\sqrt[4]{4x}})<-1,x>0$
$\Rightarrow (1+\frac{1}{\sqrt[4]{4x}})>{\left(\frac{1}{2}\right)}^{-1},x>0$
$\Rightarrow 1+\frac{1}{\sqrt[4]{4x}}>2,x>0$
$\Rightarrow \frac{1}{\sqrt[4]{4x}}>1\Rightarrow x^{\frac{1}{4}}<1,x>0\Rightarrow 0<x<1,x>0$
$\therefore D\left(f\right)=(0,1)$