One example of a relation rule that is not a function, but whose inverse is a function, is the relation rule that maps a set of x-coordinates to a set of y-coordinates, where multiple x-coordinates map to the same y-coordinate.
For instance, let's consider the relation rule:
R = {(1, 2), (2, 3), (3, 4), (1, 5)}
In this relation rule, we can see that the x-coordinate 1 maps to both y-coordinates 2 and 5. This violates the definition of a function, which requires each x-coordinate to map to only one y-coordinate.
However, if we consider the inverse of this relation rule, where we switch the x- and y-coordinates, we obtain:
R^(-1) = {(2, 1), (3, 2), (4, 3), (5, 1)}
The inverse relation rule, R^(-1), is a function because each y-coordinate now maps to only one x-coordinate.
I hope this helps! ;)