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Inverse (functions)


In mathematics, an inverse function (or anti-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: XY 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: YX will exist having the necessary relationship with f.

Let f be a function whose domain is the set X, and whose image (range) is the set Y. Then f is invertible if there exists a function g with domain Y and image X, with the property:


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