Let there be a r.v. $X$, a function $g$ that is monotone only in the interval defined in the density support and inverse denoted by $g^{-1}$. Is $\mathbb{P}(g(X)\leq a) = P(X \leq g^{-1}(a))?$
At first I was thinking specifically about the absolute value, $\lvert x\rvert$, and the r.v. $X\sim U(0,1)$, but then the doubt ocurred in a more general way. I'd like some insight in both the general and the specific cases. Thanks in advance!