Generalized inverse function for $Y=\min(X, b)$

Question

$X$ is a continuous and non-negative random valuable with CDF $F$, $b>0$, what is the generalized inverse function for $Y=\min(X, b)$

Answer 1

$$X: \Omega \rightarrow \mathbb R^+$$

$$Y = f \circ X$$ where for $b \in \mathbb R^+$ as a constant and $x \in \mathbb R$ $$f(x) = \min(x, b)$$

Then

$$Y^{-1} = X^{-1} \circ f^{-1}$$

Notice: $f^{-1}$ does not exist for $x = b$, but you could map it back to a set in $\Omega$, instead of a point in $\Omega$, i.e. the set $X^{-1}(\{x \ge b \}) \subset \Omega$.