Suppose I have a function $f: X \to Y$, and I choose some subset $Y' \subset Y$. Is there a name for the set $X'$ such that for some element $e$, $e \in X'$ if and only if $f(e) \in Y'$? For example, if $f(x) = x^2$ is defined on the reals (i.e. $X,Y = \mathbb{R}$), and $Y' = \{4, 9\}$, then $X' = \{-2, 2, -3, 3\}$.
Thanks!
$X'$, the set of elements mapped to $Y'$ by $f$, is called the pre-image or inverse image of $Y'$ under $f.$
It can be denoted $f^{-1}[Y']$. When $Y'$ is a singleton, $f^{-1}[Y']$ is also called the fiber.