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injective is surjective

Source: Romania District Olympiad 2013,grade X(problem 3)

March 14, 2013
functionalgebra proposedalgebra

Problem Statement

Take the function f:R→Rf:\mathbb{R}\to \mathbb{R}, f(x)=ax,x∈Q,f(x)=bx,x∈R\Qf\left( x \right)=ax,x\in \mathbb{Q},f\left( x \right)=bx,x\in \mathbb{R}\backslash \mathbb{Q}, where aa and bb are two real numbers different from 0. Prove that ff is injective if and only if ff is surjective.
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