If $f:A \to B$ is a real function, and $y\in B, $ Is Rudin's definition of $f^{-1}(y)$ commonly accepted as convention or not?

Question

Let $f:A \to B\ $ be a function.

In Rudin's PMA, at the bottom of page 24 and top of page 25, he states:

If $y \in B, f^{-1}(y)\ $ is the set of all $x \in A\ $ such that $f(x) = y.\ $

This notation could be confused with the function $f^{-1}:B \to A.$

Is Rudin's use of $f^{-1}(y)$ as a set still conventional today?

Answer 1

The notation $f^{-1}(\{y\}) = \{x\in A : f(x) \in \{y\}\}$ or more generally for a subset $C\subseteq A$ $$ f^{-1}(C) = \{x\in A : f(x) \in C\} $$ is pretty standard and known as the preimage.

When it is clear by context $f^{-1}(y)$ would also work but i would always make sure it is clear [as it might lead to confusion with the inverse function $f^{-1}$ which not necessarily exists]

Answer 2

Yes, $f^{-1}(y)$ is a common (I dare say the most common) notation for the set $f^{-1}(\{y\})$, when it is clear by context that $f^{-1}$ does not indicate a purported inverse map of $f$.