Criterion for dense image of a morphism between affine schemes

Question

My question:
Let $A$ and $B$ be finitely generated commutative $k$-algebras, and both be integral domain (k is a field). Let $g: A\longrightarrow B$ be a homomorphism of $k$-algebra, $f=g^{*}: Y = \mathrm{Spec}(B) \longrightarrow X = \mathrm{Spec}(A)$ is the mapping of prime spectral spaces. Let $\dim(Y)=\dim(X)$, and there exists a closed point $x$ of $X$ such that $f^{-1}(x)$ is a non-empty finite set. Claim: $f(Y)$ is dense in $X$.
My attempt:
If $F$ is an irreducible branch of $f^{-1}(x)$, then $\dim(F)\geq 0$. But I don't know what to do next. Using the method of proof to the contrary cannot handle $X-\overline{f(Y)}$.
I am a beginner in commutative algebra, and this is a homework question for me to study Kurll dimension theory. Therefore, I hope to provide a relatively basic proof method.