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