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

For f to be defined we must have ${\log }_{1/2}\left(1+\frac{1}{\sqrt[4]{x}}\right)<-1\iff 1+\frac{1}{\sqrt[4]{x}}>{\left(2^{-1}\right)}^{-1}=2$

which is possible if and only if $\frac{1}{\sqrt[4]{x}}>1$i.e. 0<x<1. Hence the domain of the given function is  $\left\{x:0<x<1\right\}$