inverseable

Inverseable is a term used in mathematics to describe a function that has an inverse function. A function f is inverseable if and only if there exists a function g such that for all x in the domain of f, g(f(x)) = x, and for all y in the domain of g, f(g(y)) = y. The inverse function is often denoted as f⁻¹. Geometrically, a function is inverseable if and only if it is bijective, meaning it is both injective (one-to-one) and surjective (onto). Injectivity ensures that each output value corresponds to a unique input value, preventing ambiguity when trying to find the original input. Surjectivity guarantees that every possible output value in the codomain can be reached by the function. If a function is not bijective, it may still be possible to restrict its domain and codomain to make it inverseable. For instance, the function f(x) = x² is not inverseable over all real numbers because it is not injective (both 2 and -2 map to 4). However, if we restrict the domain to non-negative real numbers, then f(x) = x² becomes inverseable with the inverse function g(x) = √x. The concept of inverseable functions is fundamental in many areas of mathematics, including algebra, calculus, and linear algebra.