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 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:

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, e.g. arcsin x for the inverse of sin(x).

Simplifying rule

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.

Indeed, an alternative definition of an inverse function g of f is to require that g o f resp. f o g be the identity function on the domain resp. codomain of f.

Existence

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 y = k 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

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.

The symbol f ^-1 is also used for the (set valued) function associating to an element or a subset of the codomain, the inverse image of this subset (or element, seen as a singleton).

See also

• implicit function theorem
• inverse image