Let $k$ be a field, $R = k[x_1,...,x_n]$ and $\Phi: R \rightarrow R$ be an endomorphism. It is known that if $\Phi$ is surjective then it is also injective (see http://mathoverflow.net/questions/71185/a-proof-for-a-statement-about-polynomial-automorphism).
For the inverse problem, it is known that if $k$ is a finite field or an algebraically closed field then $\Phi$ is injective implies $\Phi$ is surjective (see http://terrytao.wordpress.com/2009/03/07/infinite-fields-finite-fields-and-the-ax-grothendieck-theorem/ for the proof when $k$ is finite or $\mathbb{C}$).
The remaining question is an counter-example in which $k$ is not finite nor algebraically closed (let $k = \mathbb{R}$, for instance), $\Phi$ is injective but not surjective.
