rightinverse

A right inverse of a function f: A → B is a function g: B → A such that f(g(b)) = b for every b in B. In category-theoretic terms, g is a section of f and f ∘ g equals the identity on B.

A function f has a right inverse if and only if f is surjective. If f is surjective, a right inverse can be defined by choosing, for each b in B, some a in A with f(a) = b; the axiom of choice is typically invoked to make such selections in general. If a right inverse g exists, then f is surjective by construction.

The existence of a right inverse implies f is not necessarily injective, but it guarantees a way to recover any element of B from some element of A. If f also has a left inverse h with h ∘ f = id_A, then f is bijective, and the left and right inverses coincide with the inverse function of f.

In linear algebra, for a linear map T: V → W, a right inverse exists precisely when T is surjective; equivalently, there is a linear map S: W → V with T ∘ S = id_W. The existence of such a linear right inverse is tied to whether the short exact sequence 0 → Ker(T) → V → W → 0 splits.

Examples illustrate these ideas: the function f(x) = x^3 from R to R has a right inverse g(y) = ∛y since f(g(y)) = y. A non-surjective example is f: {a,b} → {1,2} defined by f(a) = 1, f(b) = 1; since 2 is not attained, no right inverse exists.