I was recently introduced to a rule for inverse functions that specifies that an ascending function and its inverse will only intersect on the $y=x$ axis if they intersect at all, while descending functions and their inverses can intersect at infinite points.
Well, below we have the familiar $\frac{1}{x}$ graph with the $y=x$ axis:
So as we see, it has infinite intersection points with its inverse, but this is normal because it is a descending function. Now let's take a look at $\frac{-1}{x}$:
$\frac{-1}{x}$ is ascending, but it seems to also have infinite points of intersection with its inverse.
Am I making some kind of obvious mistake or is $\frac{-1}{x}$ really breaking the aforementioned rule?
The rule is correct. The function $f(x)=-1/x$ is neither ascending nor descending on its entire domain.
Comment: At first I thought the domain of the function had to be an interval (i.e., no gaps) for the rule to apply, but this is not necessary.