Preimages

Preimage, or inverse image, of a set under a function is the set of all elements in the domain that map into that set. If f: X → Y and A ⊆ Y, then the preimage of A is f^{-1}(A) = { x ∈ X | f(x) ∈ A }. For a single element y ∈ Y, f^{-1}({y}) = { x ∈ X | f(x) = y } is called the fiber over y.

Notation and special cases: The symbol f^{-1}(A) denotes the preimage of a subset A of Y. The phrase inverse function is only appropriate when f is bijective; then f^{-1}(y) is the unique x with f(x) = y, and f^{-1} acts as an actual function from Y to X.

Basic properties: Preimages preserve set operations: f^{-1}(∪_i A_i) = ∪_i f^{-1}(A_i) and f^{-1}(∩_i A_i) = ∩_i f^{-1}(A_i). They also satisfy f^{-1}(Y \ A) = X \ f^{-1}(A). In topology, if f is continuous, the preimage of every open set is open, which characterizes continuity.

Examples: Let f: R → R be f(x) = x^2 and A = [1, 4]. Then f^{-1}(A) = { x ∈ R | x^2 ∈ [1, 4] } = (-2, -1] ∪ [1, 2). The preimage of a single point y is the set of all x with f(x) = y, which may be one or more points.

Applications: Preimages are used to describe solutions to f(x) ∈ A, to analyze fibers, and in measure and probability theory to pull back sets along functions.