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.

A less trivial offering is

f(x) = x

g(x) = 1/x

Note that

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

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

This works as a problem solution if we limit the domain of both to x not in (-1,0,1)

Or, if you prefer, we could transform g(x) into

g(x) = (x^2 - 1)/(x^3 - x). Arguably, this equals 1/x except for x in (-1,0,1), where it is undefined.

Similarly, f(x) = (x^4 - x^2)/(x^3 - x).

Arguably, this equals x except for x in (-1,0,1), where it is undefined.

So, one answer is

f(x) = (x^4 - x^2)/(x^3 - x)

g(x) = (x^2 - 1)/(x^3 - x)

and this seems to meet all puzzle conditions.

It also works if they are reversed:

g(x) = (x^4 - x^2)/(x^3 - x)

f(x) = (x^2 - 1)/(x^3 - x)