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

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

$$X: \Omega \rightarrow \mathbb R^+$$
$$Y = f \circ X$$ where for $$b \in \mathbb R^+$$ as a constant and $$x \in \mathbb R$$ $$f(x) = \min(x, b)$$
$$Y^{-1} = X^{-1} \circ f^{-1}$$
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 $$X^{-1}(\{x \ge b \}) \subset \Omega$$.