The binary reading of the general solution

Write ‾b_m … b₀‾₂ for the binary representation of n, and extend the notation to arbitrary families of digits, not necessarily equal to 0 or 1. Show that f(‾1 b_{α−1} … b₀‾₂) = ‾γ β_{b_{α−1}} … β_{b₁} β_{b₀}‾₂. (Question IV.3 of the assignment.)