Solve y=(-sqrt{2-1/(3-x)^-1/2}+1)^1/2

$$y = ( - \sqrt { 2 - 1 / ( 3 - x ) ^ { - 1 / 2 } } + 1 ) ^ { 1 / 2 }$$

Solve for x (complex solution)

$$\left\{\begin{matrix}x=-\left(1+2y^{2}-y^{4}\right)^{2}+3\text{, }&y\neq -\sqrt{\sqrt{2}+1}\text{ and }y\neq i\sqrt{\sqrt{2}-1}\text{ and }y\neq -i\sqrt{\sqrt{2}-1}\text{ and }arg(1+2y^{2}-y^{4})<\pi \text{ and }arg(1-y^{2})<\pi \text{ and }\left(y=0\text{ or }arg(y)<\pi \right)\text{ and }y\neq \sqrt{\sqrt{2}+1}\\x=-1\text{, }&y=1\end{matrix}\right.$$

Solve for y (complex solution)

$$y=\sqrt{-\sqrt{-\sqrt{3-x}+2}+1}$$

$$x\neq 3$$

Solve for x

$$x=-\left(1+2y^{2}-y^{4}\right)^{2}+3$$

$$1+2y^{2}-y^{4}\geq 0\text{ and }1-y^{2}\geq 0\text{ and }y\geq 0\text{ and }y\leq \sqrt{3}\text{ and }\left(y\leq \sqrt{2}\text{ or }y\geq \sqrt{\sqrt{3}+1}\right)$$

Solve for y

$$y=\sqrt{-\sqrt{-\sqrt{3-x}+2}+1}$$

$$x\geq -1\text{ and }x\leq 2$$