I'm starting to think that \$$g\$$ is in fact surjective onto \$$\mathbb{Q}\$$. I tried to prove it using \$$f^{-1}(x) = 2\lceil 1/x-1 \rceil - (1/x-1) \$$, which can be understood by noting that \$$2\lfloor x \rfloor - x \$$ "flips" \$$x\$$ over it's floor (ex: flips \$$\frac{5}{4}\$$ over \$$\frac{4}{4}\$$ to \$$\frac{3}{4}\$$) so \$$2\lceil y \rceil - y \$$ reverses by "flipping" \$$y\$$ over it's ceil, but haven't been successful without a good definition of "simplest nonnegative rational": https://www.wolframcloud.com/objects/14fd39f3-1c50-4361-8bf7-5957ad3c4686