Functions, Relations & Transformations · Hard problem

Let $f(x) = \begin{cases} ax + 1, & x < 2 \\ a^2 x - 3, & x \ge 2, \end{cases}$ where $a$ is a constant. For which value of $a$ is $f$ one-to-one with range equal to all real numbers (so that $f^{-1}$ is defined for every real number)?