If x from a random variable $X \in R $ with a continuous distribution $F$ and $u \in [0,1]$. Then general inverse function is defined as
$$ F^{-}(u) = inf \{ x; F(x) \geq u \} $$
There is any simply explanation for this general inverse function. So that students from other discipline can understand this definition?
I will denote the function by $\Phi$ and make some remarks.
1) It is better to go for $u\in(0,1)$.
This because $\Phi(u)\in\mathbb R$ for every $u\in(0,1)$ which is not necessarily true if $u=0$ or $u=1$.
2) It is not necessary to demand that $F$ is continuous.
3) Characteristic is the relation:$$F(x)\geq u\iff x\geq \Phi(u)$$Based on that relation it can be shown that random variable $\Phi(U)$ - where $U$ has uniform distribution on $(0,1)$ - will have $F$ as CDF.
4) $\Phi$ can be recognized as a weak inverse of $F$. We have:$$F\circ\Phi\circ F=F\text{ and }\Phi\circ F\circ\Phi=\Phi$$