This is exercise 8.3 B in Vakil's FOAG.
Suppose $\pi: X \to Y$ is a quasicompact morphism of locally Noetherian scheme. Suppose $Z$ is the scheme-theoretic image of $\pi$. Then
$$\text{Ass}\ Z \subset \text{im} \ \text{Ass} \ X$$
where $\text{Ass} \ X$ refers to the set of associated points of the scheme $X$.
Following the accompanying hint, I have shown the result (without using quasicompactness) when both $X$ and $Y$ are affine using the characterization of associated points of Noetherian affine schemes $\text{Spec} \ A$ as minimal primes of $A$. I also know that quasicompact morphisms allow for the construction of the scheme theoretic image affine by affine, but this is also true when $X$ is reduced.
My questions: How do I rigorously reduce to the case when $X$ and $Y$ are affine? Does the statement still hold true if $X$ is reduced but $\pi$ is not quasicompact? What does this statement mean geometrically, and how can I make it "obvious" intuitively?