inversa

Inversa, or inverse, is a term used in mathematics and related disciplines to denote an object or operation that reverses the effect of another. In many languages, including Spanish and Portuguese, inversa is the feminine form of inverse.

Inverse function: If f: X -> Y is a bijection, there exists a unique inverse function f^{-1}: Y -> X such that f^{-1}(f(x)) = x for all x in X and f(f^{-1}(y)) = y for all y in Y. The inverse is defined only when f is bijective or when the domain is restricted to make it bijective.

Inverse matrix: For a square matrix A, an inverse A^{-1} exists if det(A) ≠ 0; then AA^{-1} = A^{-1}A = I, where I is the identity matrix. It can be computed via the adjugate formula, row reduction, or other standard methods.

Inverse element: In an algebraic structure with an identity element, an element g has an inverse g^{-1} satisfying gg^{-1} = g^{-1}g = e, where e is the identity.

Inverse relation and inverse image: If R is a relation, the inverse relation R^{-1} contains pairs (y, x) whenever (x, y) ∈ R. For a function f, the inverse image of a set B is f^{-1}(B) = {x | f(x) ∈ B}, defined even if f is not invertible.

Inverse trigonometric functions: arcsin, arccos, and arctan are the inverse functions of sin, cos, and tan on their principal domains.

Inverse operations: The opposites of common operations, such as subtraction being the inverse of addition or division the inverse of multiplication.

Etymology: from Latin inversus meaning turned back.