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Preimage

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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 $f^{-1}$.