Tim had a very thorough answer. Good job!
I'd like to add one more remark. Not every monotonically increasing function has an inverse function. Actually only strictly monotonically increasing/decreasing functions have inverse functions.
For monotonically increasing cdf which are not strictly monotonically increasing, we have a quantile function which is also called the inverse cumulative distribution function. You can find more details here.
Both inverse functions (for those strictly increasing cdf) and quantile functions (for those monotonically increasing but not strictly monotonically increasing cdfs) can be denoted as $F^{-1}$, which can be confusing sometimes.
