The domain of the following function is

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

$\begin{array}{l}f\left(x\right) \mathrm{is} \mathrm{defined} \mathrm{if}\\ -{\log }_{1/2}\left(1+\frac{1}{x^{1/4}}\right)-1>0\\ \mathrm{or} {\log }_{1/2}\left(1+\frac{1}{x^{1/4}}\right)<-1\\ \mathrm{or} 1+\frac{1}{x^{1/4}}>{\left(\frac{1}{2}\right)}^{-1}\\ \mathrm{or} \frac{1}{x^{1/4}}>1\\ \mathrm{or} 0<x<1\end{array}$