Dominant Morphism on Affine Varieties

Question

Let $X,Y\in \mathbb{A}^{n}_{k}$ affine varieties, I know that a morphism $f:X\rightarrow Y$ is dominant iff the correspondent morphism $\phi:k[Y]\rightarrow k[Y]$ is injective. How can I show from here that $\dim X$ is the largest number $n$ such that exists a dominant morphism $X\rightarrow \mathbb{A}_{k}^{n}$?