An inverse operation or function reverses the effect of another operation or function.
Addition and subtraction are inverse operations. Multiplication and division are inverses. The inverse of a function f, denoted f⁻¹, satisfies f⁻¹(f(x)) = x. A function must be one-to-one to have an inverse. The additive inverse of a is -a (a + (-a) = 0). The multiplicative inverse of a is 1/a (a × 1/a = 1). Inverses are fundamental in solving equations, matrix algebra, and understanding function behavior.
The concept of inverse functions emerged with the development of logarithms as inverses of exponentials by John Napier (1614). The notation f⁻¹ for inverse functions was introduced by John Herschel in 1813. Inverse trigonometric functions were studied by Euler. The inverse of a matrix was developed by Arthur Cayley in the 19th century.