Given the following definitions
$$ y(x) = \left\lfloor{ (x - n + a - b)/a }\right\rfloor ,\quad a,b \in \mathbb{R}, n \in \mathbb{N}, $$ $$ z(x) = \max (0, y(x)), $$ where $x, y(x), z(x) \in \mathbb{N}$ are positive integer variables and $a,b,n$ are constants.
I am trying to find an expression for $x(z)$. Somehow, I arrived at the following expression
$$ x(z) = \begin{cases} \max (0,\left\lceil{ a(z-1) + n + b }\right\rceil), & \text{if $x > 0$} \\ 0, & \text{else} \end{cases} $$
which works for $a <= 1$. I must have made a mistake when inverting, since for $a > 1$ this formula is wrong.
Any help would be appreciated!
Edit: A lower and upper bound for the inverse would also be pertinent for my application. The expression I found seems to be an upper bound for $a <= 1$, however I didn't prove it thoroughly.