Pöördfunktsioon

Pöördfunktsioon, also known as the inverse function, is a fundamental concept in mathematics. For a given function f, its inverse function, denoted as f⁻¹, essentially "reverses" the action of f. If f maps an input x to an output y, then the inverse function f⁻¹ maps the output y back to the original input x. This relationship can be expressed as f(f⁻¹(y)) = y and f⁻¹(f(x)) = x, for all valid inputs.

Not all functions have an inverse. A function must be bijective, meaning it is both injective (one-to-one) and surjective (onto), to possess an inverse. Injectivity ensures that each output corresponds to a unique input, preventing ambiguity when reversing the mapping. Surjectivity guarantees that every element in the codomain is an output of some input.

Graphically, the graph of an inverse function f⁻¹ is a reflection of the graph of the original function f across the line y = x. This is because the roles of the x and y coordinates are swapped between a function and its inverse.

Finding the inverse of a function typically involves a systematic process. One common method is to replace f(x) with y, swap the variables x and y, and then solve the resulting equation for y. The expression obtained for y will be the inverse function, f⁻¹(x).

Pöördfunktsioonid are crucial in various areas of mathematics and its applications, including solving equations, calculus, and trigonometry, where inverse trigonometric functions play a vital role.