  The main result of this article establishes the free analog of Grothendieck's
Theorem on bijective polynomial mappings of $\mathbb{C}^g$. Namely, we show if
$p$ is a polynomial mapping in $g$ freely non-commuting variables sending
$g$-tuples of matrices (of the same size) to $g$-tuple of matrices (of the same
size) that is injective, then it has a free polynomial inverse.
  Other results include an algorithm that tests if a free polynomial mapping
$p$ has a polynomial inverse (equivalently is injective; equivalently is
bijective). Further, a class of free algebraic functions, called hyporational,
lying strictly between the free rational functions and the free algebraic
functions are identified. They play a significant role in the proof of the main
result.
