US-TST 2004 Problem 6

Define the function $f: \mathbb N \cup \{0\} \to \mathbb{Q}$ as follows: $f(0) = 0$ and $$f(3n+k) = -\frac{3f(n)}{2} + k ,$$ for $k = 0, 1, 2$ . Show that $f$ is one-to-one and determine the range of $f$ .