This site is devoted to mathematics and its applications. Created and run by Peter Saveliev.

# Preimage

Given a function $f: X → Y$ and a point $b$ in $Y$, the *preimage* of $b$ is a subset of $X$:
$$f^{-1}(b) = \{x: f(x) =b\}. $$
An alternative terminology is that $f^{-1}(b)$ are the *fibers* of $f$.

If $f$ is invertible, then the above notation makes sense as the *value* of $b$ under the inverse $f^{-1}$ of $f$.

More broadly, we consider the *preimage of a subset* $B$ in $Y$:
$$f^{-1}(B) = \{x: f(x) ∈ B\}. $$

If $f$ is invertible, then the above notation makes sense as the *image* of $B$ under the inverse $f^{-1}$.