Distribution of the Square of a Standard Normal via Change of Variables

Let $X \sim N(0, 1)$.

(a) The transformation $Y = X^2$ is not monotone. Using the CDF method, derive $f_Y(y)$ by first computing $F_Y(y) = P(X^2 \leq y)$ and then differentiating.

(b) Alternatively, apply the non-monotone change-of-variables formula: if $Y = g(X)$ and $g$ has exactly two branches $x_1(y)$ and $x_2(y)$ satisfying $g(x_i) = y$, then $$f_Y(y) = \sum_{i=1}^{2} f_X(x_i(y))\, \left|\frac{dx_i}{dy}\right|.$$ Verify you get the same answer.

(c) Identify the resulting distribution and express it as a Gamma distribution.