préimagem

In mathematics, the préimage, also known as the inverse image or counterimage, of a set under a function is the set of all elements in the domain that map to elements in a given subset of the codomain. If f is a function from set A to set B, and S is a subset of B, then the préimage of S under f, denoted as f⁻¹(S), is the set {x ∈ A | f(x) ∈ S}.

It is important to note that the préimage is a set, and it may be empty even if the function itself is defined for all elements in its domain. The notation f⁻¹ is used for the préimage, but this should not be confused with the inverse function. An inverse function f⁻¹ exists only if the function f is bijective (both injective and surjective). If an inverse function exists, then the préimage of a set S under f is identical to the image of S under the inverse function, i.e., f⁻¹(S) = f⁻¹(S).

For example, consider the function f(x) = x² which maps real numbers to real numbers. The préimage of the set {4, 9} under f is the set {-3, 3, -2, 2}. This is because f(-3) = 9, f(3) = 9, f(-2) = 4, and f(2) = 4.