käänteistoiminto

Käänteistoiminto, known in English as inverse function, is a fundamental concept in mathematics that describes a function that "undoes" the effect of another function. If a function f maps an element x from a set A to an element y in a set B, then its inverse function, denoted as f⁻¹, maps y back to x. Essentially, applying a function and then its inverse, or vice versa, returns the original input.

For an inverse function to exist, the original function must be bijective, meaning it is both injective (one-to-one) and surjective (onto). Injective means that each output value corresponds to a unique input value. Surjective means that every element in the codomain (the set of possible output values) is mapped to by at least one element in the domain.

The notation f⁻¹(y) = x signifies that the inverse function f⁻¹ applied to y results in x, given that f(x) = y. Graphically, the graph of an inverse function is a reflection of the original function's graph across the line y = x. This property arises directly from the swapping of the x and y coordinates when defining the inverse. Understanding käänteistoiminto is crucial for solving equations, simplifying expressions, and comprehending more advanced mathematical topics.