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f(2x) = 2f(x), f(4x+1) = 4f(x) + 3, f(4x-1) = 2f(2x -1)-1

Source: Nordic Mathematical Contest 1997 #4

October 4, 2017
algebrainjective functionfunctional equation

Problem Statement

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