Answer
$$y=\sqrt{-\frac{x^2-1}{2}},\:y=-\sqrt{-\frac{x^2-1}{2}}$$
Work Step by Step
Switching the elements of the domain and the range (in other words, the dependent and independent variables), we find that the inverse is:
$$x=\sqrt{1-2y^2} \\ y=\sqrt{-\frac{x^2-1}{2}},\:y=-\sqrt{-\frac{x^2-1}{2}}$$