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Unchanged functions after transformation

Source: Canadian Mathematical Olympiad - 1983 - Problem 2.

June 26, 2011
functionlogarithmsalgebrafunctional equationalgebra unsolved

Problem Statement

For each r∈Rr\in\mathbb{R} let TrT_r be the transformation of the plane that takes the point (x,y)(x, y) into the point (2rx;r2rx+2ry)(2^r x; r2^r x+2^r y). Let FF be the family of all such transformations (i.e. F={Tr:r∈R}F = \{T_r : r\in\mathbb{R}\}). Find all curves y=f(x)y = f(x) whose graphs remain unchanged by every transformation in FF.
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