MathNet 1997 — Part 01 Question 78

Let $f$ be a function defined in the set $\{0,1,2, \ldots\}$ of non-negative integers, satisfying $f(2x) = 2f(x)$, $f(4x+1) = 4f(x) + 3$, and $f(4x-1) = 2f(2x-1) - 1$. Show that $f$ is an injection, i.e. if $f(x) = f(y)$, then $x = y$.