Finding the distribution of $f_{Y|X=x}$ and $f_{X|Y=y}$

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$X$ and $Y$ are random variables. $X$ has uniform distribution in $[-1,1]$, i.e, $F_{X} = 1/2$ in $[-1,1]$, and 0c.c. $Y = X^{2}$

What are the distribution of $f_{Y|X=x}$ and $f_{X|Y=y}$?

I found that the marginal o Y is: $f_{Y} = 1/2\sqrt{y}$. Am I right?

And then I, just stuck here: $f_{Y|X=x} = f_{x,y}/f_{x} = 2 f_{x,y}$

What can I do after this?

Any help?