As the transformation of y=f^-1 is a reflection in the line y=x of the original function, it seems like it is possible to find the points of intersection between the two functions by solving y=f(x) for when y=x (would this guarantee to find every point of intersection?).

Alternatively, it also seems possible to just equate and solve for when f(x)=f^-1

Are there any other ways/tricks to find the points of intersection?

Also, are there any ways of finding the number of times (if any) that a function intersects with its inverse without finding all points individually?

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