$\begin{matrix} {{D}_{f}}=\left\{ x\left| 4 \right.-x\gt 0 \right\}=(-\infty ,4) & {}  \\ \end{matrix}$              ${{D}_{g}}=\mathbb{R}$ ${{D}_{fog}}=\left\{ x\in {{D}_{g}}\left| g(x)\in {{D}_{f}} \right. \right\}=\left\{ x\in \mathbb{R}\left| {{x}^{2}}+1\lt 4 \right. \right\}$ ${{D}_{fog}}=\left\{ x\in \mathbb{R}\left| {{x}^{2}}\lt 3 \right. \right\}\Rightarrow {{D}_{fog}}=(-\sqrt{3},\sqrt{3})$ ${{D}_{gof}}=\left\{ x\in {{D}_{f}}\left| f(x)\notin {{D}_{g}} \right. \right\}=\left\{ x\lt 4\left| \frac{1}{\sqrt{4-x}} \right.\in \mathbb{R} \right\}=(-\infty ,4)$ ${{D}_{fog}}\bigcap {{D}_{gof}}=(-\sqrt{3},\sqrt{3})\bigcap (-\infty ,4)=(-\sqrt{3},\sqrt{3})$