Injective Function - Injections Can Be Undone

Injections Can Be Undone

Functions with left inverses are always injections. That is, given f : X → Y, if there is a function g : Y → X such that, for every x ∈ X

g(f(x)) = x (f can be undone by g)

then f is injective. In this case, f is called a section of g and g is called a retraction of f.

Conversely, every injection f with non-empty domain has a left inverse g (in conventional mathematics). Note that g may not be a complete inverse of f because the composition in the other order, f g, may not be the identity on Y. In other words, a function that can be undone or "reversed", such as f, is not necessarily invertible (bijective). Injections are "reversible" but not always invertible.

Although it is impossible to reverse a non-injective (and therefore information-losing) function, one can at least obtain a "quasi-inverse" of it, that is a multiple-valued function.