CMO 1983 Problem 2

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