$F[x]/(x)^n\simeq F[y]/(ay+b)^n$? Let $F$ be a field. Then $F[x]/(x)^n\simeq F[y]/(ay+b)^n$ for $a,b\in F\setminus\{0\}$

This is a question related to this question. Is the statement true? I can't apply the method mentioned in the comment there. Could you give any hints?
This kind of questions is sometimes confusing for beginners, as they use the letter $x$ on both sides.
It would be much clearer if the question is stated like this:

Show that $F[x]/(x)^n \simeq F[y]/(ay + b)^n$.

It is then easy to construct homomorphisms in both directions.
$\rightarrow$: send $x$ to $ay + b$.
$\leftarrow$: send $y$ to $a^{-1}(x - b)$.
And then easy to check that they are well-defined and are inverses of each other.
This kind of questions is sometimes confusing for beginners, as they use the letter $x$ on both sides.
It would be much clearer if the question is stated like this:

Show that $F[x]/(x)^n \simeq F[y]/(ay + b)^n$.

It is then easy to construct homomorphisms in both directions.
$\rightarrow$: send $x$ to $ay + b$.
$\leftarrow$: send $y$ to $a^{-1}(x - b)$.
And then easy to check that they are well-defined and are inverses of each other.