Well, turns out there is such a function.
y(x) = 0.743*x^(1.618)
I took a more formal approach to the problem that didn't get me anywhere, so I talked to a buddy and here's what we found.
Explanation.
As you see, nothing formal. A mathematician will say it isn't a proof at all, but it does show that there's at least one said function.
PS - My statement on a previous post that y=x^3 was not bijective is wrong.