Inverse function

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:

f ^-1(f(x)) = f(f ^-1(x)) = x.

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

f : x → 3x + 2

f ^-1 : x → (x - 2) / 3

The superscript "-1" is not an exponent. Similarly, f ²(x) means "do f twice", that is f(f(x)), not the square of f(x) (unfortunately, this notation has an exception for the trigonometric functions: 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:

• each element in the codomain must be "hit" by f: otherwise there would be no way of defining the inverse of f for some elements
• each element in the codomain must be "hit" by f only once: otherwise the inverse function would have to send that element back to more than one value.

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

y = f(x)

for x, and then exchanging y and x to get

y = f ^-1(x).

This is not always easy; if the function f(x) is analytic, the Lagrange inversion theorem may be used.