Derive distribution of an inverse transform

Suppose $X$ has density $f_X$ on $(0,1)$, and let $g(u) = \frac{1}{1 + e^{-u}}$ be the strictly increasing logistic map. (A) Derive the general formula for the density of $U = g^{-1}(X)$ in terms of $f_X$ and $g$. (B) Take $X \sim \mathrm{Uniform}(0,1)$ and compute the distribution, mean, and variance of $U = \log\left(\frac{X}{1-X}\right)$. Show the Jacobian steps and the support clearly. Overview: This question tests change-of-variables and inverse-transform techniques in probability, focusing on the logit (inverse logistic) mapping, Jacobian-based density transformation, handling of support, and computation of distributional moments.