Inverse operation - Definition and Overview

In mathematics, 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 <math>x \in X<math> we have:

<math>f^{-1}(f(x))=f(f^{-1}(x))=x<math>.

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

<math>f\colon x\to 3x+2<math>

<math>f^{-1}\colon x\to(x-2)/3<math>

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:

• 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.

If f is a real-valued function, then for f to have a valid inverse, it must pass the horizontal line test, that is a horizontal line <math>y=k<math> placed on the graph of f must pass through f exactly once for all real k.

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

<math>y=f(x)<math>

for x, and then exchanging y and x to get

<math>y=f^{-1}(x)<math>

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