I don't know how to invert functions like this. $\displaystyle \begin{cases}\frac{-x^2}{2}+\frac{1}{2} & x\leq 0 \\ \frac{x^2}{2}+\frac{1}{2} & x> 0 \end{cases}$
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make a quick sketch of it and solve for the inverse for $y<\frac{1}{2}$ and for $y\geq \frac{1}{2}$ separately.
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