$X\sim U(-1,3)$, and $Y\sim X^4$. How to find the p.d.f of $Y$ using transformation of variable?
This is not a duplicated question of this. I would like to know how to use transformation of variable method to step-by-step derive the p.d.f of $Y$.
For transformation of variable method, I mean $$f_Y(y) = f_X(x(y))|\frac{dx}{dy}|$$
Here is what I have attempted.
Since $Y=X^4$, we have
$$X=\begin{cases}-Y^{\frac{1}{4}}&, X\in[-1,0) \\ Y^{\frac{1}{4}}&, X\in[0,3] \end{cases}$$
Then $$|\frac{dx}{dy}|=\frac{1}{4}y^{-\frac{3}{4}}$$
Also since $f_X(x) = \frac{1}{4}\mathbb{1}_{-1\leq X<0}+\frac{1}{4}\mathbb{1}_{0\leq X\leq3}$, we have $$f_Y(y) = \frac{1}{4}\mathbb{1}_{-1\leq -Y^{\frac{1}{4}}<0}|\frac{dx}{dy}| + \frac{1}{4}\mathbb{1}_{0\leq Y^{\frac{1}{4}}\leq3}|\frac{dx}{dy}|,$$ which means
$$f_Y(y) = \frac{1}{8}y^{-\frac{3}{4}}\mathbb{1}_{0\leq Y\leq1} + \frac{1}{16}y^{-\frac{3}{4}}\mathbb{1}_{1\leq Y\leq3^4}$$
However, I don't think this is correct, and I don't know which step I made mistake. How to use transformation of variable method to step-by-step derive the p.d.f of $Y$?