The Inverse function reference article from the English Wikipedia on 24-Jul-2004
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Inverse function

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In mathematical analysis, an inverse function is in simple terms a function which "does the reverse" of a given function. More formally, if f is a function with domain X, then f -1 is its inverse function if and only if for every we have:


For example, if the function x → 3x + 2 is given, then its inverse function is x → (x - 2) / 3. This is usually written as:

The superscript "-1" is not an exponent. Similarly, as long as we are not in trigonometry, f 2(x) means "do f twice", that is f(f(x)), not the square of f(x). For example, if : f : x → 3x + 2, then f 2 : x = 3*((3x + 2)) + 2, or 9x + 8. However, in trigonometry, for historical reasons, sin2(x) usually does mean the square of sin(x). As such, the prefix arc is sometimes used to denote inverse trigonometric functions, eg arcsin x for the inverse of sin x).

Generally, if f(x) is any function, and g is its inverse, then g(f(x)) = x and f(g(x)) = x. In other words, an inverse function undoes what the original function does. In the above example, we can prove f -1 is the inverse by substituting (x - 2) / 3 into f, so

3(x - 2) / 3 + 2 = x.
Similarly this can be shown for substituting f into f -1.

For a function f to have a valid inverse, it must be a bijection, that is:

It is possible to work around this condition, by redefining f's codomain to be precisely its range, and by admitting a multi-valued function as an inverse.

If one represents the function f graphically in an x-y coordinate system, then the graph of f -1 is the reflection of the graph of f across the line y = x.

Algebraically, one computes the inverse function of f by solving the equation

for x, and then exchanging y and x to get
This is not always easy; if the function f(x) is analytic, the Lagrange inversion theorem may be used.