$\text{The domain of }f\left(x\right)=\sqrt{{\log }_{0.4}\left(\frac{x-1}{x+5}\right)}\text{ is}$

$f\left(x\right)=\sqrt{{\log }_{0.4}\left(\frac{x-1}{x+5}\right)}$

exists if
$\begin{array}{l}{\log }_{0.4}\left(\frac{x-1}{x+5}\right)\ge 0\text{ and }\left(\frac{x-1}{x+5}\right)>0\\ \text{⇒}\frac{x-1}{x+5}\le (0.4{)}^0\text{ and }\frac{x-1}{x+5}>0\\ \text{⇒}\frac{x-1}{x+5}\le 1\text{ and }\frac{x-1}{x+5}>0\\ \text{⇒}\frac{x-1}{x+5}-1\le 0\text{ and }\frac{x-1}{x+5}>0\end{array}$

$\begin{array}{l}\text{⇒}\frac{-6}{x+5}\le 0\text{ and }\frac{x-1}{x+5}>0\\ \text{⇒}x+5>0\text{ and }\frac{x-1}{x+5}>0\\ \text{⇒ }x>-5\text{ and }x-1>0(\text{ As }x+5>0)\end{array}$

$\text{ or }x>-5\text{ and }x>1$

$\text{ Thus, the domain }f\left(x\right)\text{ is }(1,\mathrm{\infty })\text{ . }$