Knowlpedia

Preimage (inverse image)

The set of inputs that a function maps into a given subset of the codomain.

Let f:X→Yf:X\to Y be a function and let B⊆YB\subseteq Y. The preimage (or inverse image) of BB under ff is

f−1(B):={x∈X:f(x)∈B}⊆X.f^{-1}(B):=\{x\in X:f(x)\in B\}\subseteq X.

The notation f−1(B)f^{-1}(B) does not require ff 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 ).

Examples
  • If f:R→Rf:\mathbb{R}\to\mathbb{R}, f(x)=x2f(x)=x^2, then f−1({1})={−1,1}f^{-1}(\{1\})=\{-1,1\}.
  • For the same ff, f−1((−∞,1])=[−1,1]f^{-1}((-\infty,1])=[-1,1].
  • If f(x)=x2f(x)=x^2, then f−1((−1,0))=∅f^{-1}((-1,0))=\varnothing since x2≥0x^2\ge 0 for all x∈Rx\in\mathbb{R}.