En matemáticas, si f es una aplicación o función que lleva elementos de I en elementos de J, en ciertas condiciones será posible definir la aplicación f -1 que realice el camino de vuelta de J a I. En ese caso diremos que f -1 es la aplicación inversa o recíproca de f.

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In mathematics, an inverse function is a function that "reverses" another function: if the function f applied to an input x gives a result of y, then applying its inverse function g to y gives the result x, and vice versa. i.e., f(x) = y if and only if g(y) = x. As a simple example, consider the real-valued function of a real variable given by f(x) = 5x − 7. Thinking of this as a step-by-step procedure (namely, take a number x, multiply it by 5, then subtract 7 from the result), to reverse this and get x back from some output value, say y, we should undo each step in reverse order. In this case that means that we should add 7 to y and then divide the result by 5. In functional notation this inverse function would be given by, With y = 5x − 7 we have that f(x) = y and g(y) = x. Not all functions have inverse functions. In order for a function f: X → Y to have an inverse, it must have the property that for every y in Y there must be one, and only one x in X so that f(x) = y. This property ensures that a function g: Y → X will exist having the necessary relationship with f.

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