Käänteisfunktion

A käänteisfunktion, also known as an inverse function, is a mathematical concept that describes a function which "reverses" another function. For a function f to have an inverse, it must be bijective, meaning it is both injective (one-to-one) and surjective (onto). This ensures that for every output of f, there is exactly one input that produces it, and every possible output is covered.

The inverse of a function f, denoted as f^-1, is defined such that for any x in the domain of f, f(f^-1(x)) = x, and for any y in the range of f, f^-1(f(y)) = y. In other words, applying f to the result of f^-1 gives the original input, and applying f^-1 to the result of f gives the original output.

Not all functions have inverses. For example, the function f(x) = x^2 is not injective because it maps both 2 and -2 to 4. However, the function f(x) = x^2 restricted to the domain [0, ∞) is injective and thus has an inverse, which is the square root function.

Inverse functions are useful in various areas of mathematics and its applications. They are used to solve equations, understand the behavior of functions, and in the study of group theory and other abstract structures.