With the change of variables-method for a many-to-one transformation function Y = t(X), what's the logic behind summing the different densities for the roots of x = t^-1(y)? Probabilities should be ok to add, but densities? Also, is there no way to extend this method for many-to-many transformation functions, like calculating the density for each of the roots of y = t(x) separately and summing those sums of densities? For example, if X has uniform distribution (-1 < x < 1) and Y = +/- sqrt(1-X^2), maybe it's possible to sum the two densities of positive/negative x for both positive and negative y, then add the two densities of positive/negative y and divide by two? Would that work if there is asymmetry?