Under what conditions is the inverse of a function a function?
Can/does this have something to do with the input and outputs of a function
or is it as simple whether or not it passes the vertical line test?
It is as simple as the vertical line test.
There's an "eyeball" approach, too.
Look at the graph of the function f(x).
If it passes the "horizontal line test", its inverse is a function.
(That is, if all horizontal lines intersect the graph in at most one point,
. . then the inverse is also a function.)