$x\le 0:y={{x}^{2}}+1\,,\,y\ge 1\Rightarrow {{x}^{2}}=y-1\xrightarrow{x\le 0}x=-\sqrt{y-1}\,,\,y\ge 1\Rightarrow {{f}^{-1}}(x)=-\sqrt{x-1}\,,\,x\ge 1$ $x\gt 0:y=1-x\Rightarrow y=1-x\lt 1\Rightarrow x=1-y\,,\,y\lt 1\,\Rightarrow {{f}^{-1}}(x)=1-x\,,\,x\lt 1$ $\Rightarrow {{f}^{-1}}(x)=\left\{ \begin{matrix}-\sqrt{x-1}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x\ge 1  \\1-x\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x\lt 1  \\\end{matrix} \right.$