Consider a Beta distributed random variable, $X \sim Beta(a,b)$ Then consider the transform $$Y = \sqrt{K\frac{X}{1-X}}$$ where $K > 0$ is a constant.
How could you go about finding the PDF of $Y$? Is it a named distribution?
It seems that you cannot apply the transformation formula, since the mapping is not strictly monotonic and doesn't have an inverse.