$\begin{align}  & f(x)=1-\sqrt{x+1}\to \,{{D}_{f}}=\left[ -1,\left. +\infty  \right) \right. \\  & {{D}_{fog}}=\left\{ x\in {{D}_{f}}\left| f(x)\in {{D}_{f}} \right. \right\} \\  & =\left\{ x\in \left[ -1,\left. +\infty  \right) \right.\left| 1-\sqrt{x+1}\in \left[ -1,\left. +\infty  \right) \right. \right. \right\} \\  & 1-\sqrt{x+1}\in \left[ -1,\left. +\infty  \right) \right.\to 1-\sqrt{x+1}\ge -1\to \sqrt{x+1}\le 2 \\  & \Rightarrow 0\le \sqrt{x+1}\le 2\Rightarrow 0\le x+1\le 4\Rightarrow -1\le x\le 3 \\ \end{align}$