Find, if possible, two functions f and g with:

f(x) ≠ g(x)

g(x) = 1/f(x)

g(f(x)) = 1/f(g(x))

for all x in the respective domains.

Here's a whole family of satisfying functions:

g(x) = a

f(x) = 1/a

This satisfies the problem conditions if a not in {-1,0,1). That still leaves an uncountably large number of solutions.

Given the difficulty rating of this problem, though, I suspect that Jer is looking for something a little less simple.