n. a function, usually written f^-1, whose domain and range are respectively the range and domain of a given function, f, and under which the image, y, of an element, x, is the element of which x was the image under the given function, that is f^-1(x) = y if and only if f(y) = x; the function whose composition with the given function is the identity function. In order that the inverse should have a unique value for each argument, and so be properly a function, the given function must be injective. For example, the extraction of positive square roots, √ x, is the inverse of squaring, x², since y = x² if and only if x = √ y, and
√(x²) = (√ x)² = x;
however, without the restriction to positive values, the square root function on the domain of real numbers does not have an inverse. See also left inverse, right inverse.