rightinverseidest

Rightinverseidest is not a standard term in mathematics. It appears to be a concatenation of the phrase “right inverse” with the Latin abbreviation id est (i.e.), or it may simply be a misspelling. In what follows, the article explains the established concept of a right inverse and notes how the unfamiliar form would typically be interpreted.

A right inverse of a function. Let f be a function from a set B to a set A. A function g from A to B is called a right inverse of f if the composition f∘g equals the identity on A, that is, f(g(a)) = a for every a in A. In this situation, g is a kind of reverse map that recovers a from its image under f. A fundamental consequence is that f is surjective (onto): every a in A has a preimage g(a) in B.

Left versus right inverses. If a function f has a left inverse h with h∘f = id_B, then f is injective (one-to-one). A function having a right inverse does not necessarily have a left inverse, and vice versa. In category theory, a morphism with a right inverse is called a split epimorphism.

Existence and examples. A surjective function f: B→A always has a right inverse when the axiom of choice is assumed, though such inverses may not be unique. Concrete examples: f: R→R, f(x) = x^3 has a right inverse g(y) = the real cube root of y, since f(g(y)) = y. A projection p: R^2 → R given by p(x,y) = x has a right inverse s(t) = (t,0), because p(s(t)) = t.

In linear algebra, a linear map T: V→W has a (linear) right inverse precisely when T is surjective; then a section S: W→V with T∘S = id_W exists and may be chosen linearly.

Note. If you meant a standard term, the commonly used notion is “right inverse.” The phrase “rightinverseidest” is not a recognized term in mainstream references.