the cdf method
When you need the distribution of Y = g(X), the cdf method is the safe, always-applicable workhorse. It never asks whether g is one-to-one, smooth, or invertible. It just translates a question about Y into a question about X using the cumulative distribution function, and then, if Y is continuous, differentiates at the end to get the density.
The recipe has three steps. First, write the cdf of Y as an event about Y: F_Y(y) = P(Y at most y). Second, replace Y by g(X) and solve g(X) at most y for X — find the set of x-values that make it true — then read off that probability from the cdf of X. Third, if you want the density of a continuous Y, differentiate: f_Y(y) = d/dy F_Y(y). The whole method is just: pull the event back to X, evaluate with what you already know, then differentiate.
Because it works for any g, the cdf method is the one to reach for when the Jacobian formula does not apply: when g is not monotone (like g(x) = x^2, which folds the line), when g is flat on stretches, or when the answer is a mixed distribution. The Jacobian formula is really just the cdf method carried out once in general for a smooth monotone g; when in doubt, fall back to the cdf method.
Let X be standard normal and Y = X^2. Since g is not one-to-one, use the cdf method: for y > 0, F_Y(y) = P(X^2 at most y) = P(-sqrt(y) at most X at most sqrt(y)) = Phi(sqrt(y)) - Phi(-sqrt(y)). Differentiating gives the chi-squared density with 1 degree of freedom. The folding at x and -x is handled automatically by the two-sided interval.
A non-monotone g like squaring is handled cleanly because both branches land inside one interval.
Mind the direction of inequalities for decreasing g: if g is decreasing, g(X) at most y means X at least g^{-1}(y), so the inequality flips. Slipping here is the commonest mistake in the method.