Dominant affine images contain a principal open

Use normalization over an open to obtain $0\mathrm{\ne }a\in A=k[Y]$ with $B_a=k[X{]}_a$ finite over the injected polynomial algebra $R=A_a[t_1,\dots ,t_r]$. The open $D_Y(a)$ is nonempty: by [F3], $a$ is a nonzero polynomial function on $Y$, so it is nonzero at some point.

Fix $y\in D_Y(a)$ and the maximal ideal $\mathfrak{n}=({\mathfrak{m}}_y,t_1,\dots ,t_r)\subset R$, whose quotient is $k$. By [A1] and [F2], lying over gives a prime $\mathfrak{q}\subset B_a$ contracting to $\mathfrak{n}$. The domain $B_a\mathrm{/}\mathfrak{q}$ is finite over $k$. Every nonzero element acts injectively on this finite-dimensional vector space, hence surjectively, so the domain is a field. Algebraic closedness forces it to be $k$. The resulting $k$-algebra homomorphism $B\to B_a\to B_a\mathrm{/}\mathfrak{q}=k$ evaluates the finitely many affine-coordinate classes at elements of $k$. All defining polynomials of $X$ vanish at this tuple because their classes are zero in $B$; the restrictions of $A=k[Y]$ evaluate at $y$ because $\mathfrak{q}\cap R=\mathfrak{n}$. Hence this tuple is a classical point $x\in X$ with $f(x)=y$. Thus $D_Y(a)$ lies in the image, including when $r=0$.