For example, let's say we invert a checker algorithm and "run it backward from YES". Does that find an arbitrary single solution? Does it find all solutions? What if there are an infinite number of solutions?
Disclosure: I was that kid. My program was a mess, but good enough for a proof of concept.
In other contexts, what is meant is g : B -> 2^A such that x is in g(f(x)) for all x and f(x') = f(x) for all x' in g(f(x)). Here, g is said to produce the pre-image under f, and is a generalization of the above definition of inverse for non-injective functions.
In other contexts still, what is meant is g : B -> A such that f(g(y)) = y for all y such that there exists some x with f(x) = y. This is a different generalization called a one-sided inverse. (People probably call it a left inverse or right inverse, but I can never remember which is which because it emerges from an arbitrary notational convention.)
The article uses inverse in this third sense. "Find some input to f which would yield a given output y, if one exists."