I have the following PDF:
\begin{equation} f_X(x) = \begin{cases} \frac{1}{2 \, \sqrt{1 - x}} &\text{if } 0 \leq x \leq 1 \\ 0 &\text{otherwise} \end{cases} \end{equation}
whose domain is $\mathbb{R}$, and I've $Y = X \, (1 - X)$.
I'd like to know if:
$$ y = g(x) = x \, (1 - x) \quad \forall \, x \in \mathbb{R} $$
or if:
\begin{equation} y = g(x) = \begin{cases} x \, (1 - x) = x - x^2 &\text{if } 0 \leq x \leq 1 \\ 0 &\text{otherwise} \end{cases} \end{equation}
or if the above two $g(x)$ are equivalent.
EDIT: I'd like to know also if the determination of the different intervals of $f(g(x))$ should be done before, during or after the writing of the $f(g(x))$ pieces.
I need an explanation on how to define the endpoints of $g(x)$.