$\text{ The domain of definition of the }$function $f\left(x\right)=\sqrt{1+{\log }_e(1-x)}$

$\begin{array}{l}f\left(x\right)=\sqrt{1+{\log }_e(1-x)}\text{ value of }f\left(x\right)\text{ is real when }\\ 1+{\log }_e(1-x)\ge 0\text{ and }1-x>0\\ \Rightarrow {\log }_e(1-x)\ge -1\text{ and }x<1\\ \Rightarrow {\log }_e(1-x)\ge -1\text{ and }x<1\\ \Rightarrow 1-x\ge \frac{1}{e}\text{ and }x<1\Rightarrow x\le \frac{e-1}{e}\text{ and }x<1\text{ . }\end{array}$