Preimage (inverse image)

Let $f:X\to Y$ be a function and let $B\subseteq Y$. The preimage (or inverse image) of $B$ under $f$ is

The notation $f^{-1}(B)$ does not require $f$ to be invertible; it is defined for every function. Preimages interact well with set operations and are central in topology (continuity via preimages of open sets) and measure theory (measurability via preimages of measurable sets).

Examples:

• If $f:\mathbb{R}\to\mathbb{R}$, $f(x)=x^2$, then $f^{-1}(\{1\})=\{-1,1\}$.
• For the same $f$, $f^{-1}((-\infty,1])=[-1,1]$.
• If $f(x)=x^2$, then $f^{-1}((-1,0))=\varnothing$ since $x^2\ge 0$ for all $x\in\mathbb{R}$.