# Inverse function

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:

For example, if the function *x* → 3*x* + 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* → 3*x* + 2, then *f* ^{ 2} : *x* = 3 ((3*x* + 2)) + 2, or 9*x* + 8. However, in trigonometry, for historical reasons, sin^{2}(*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*).

If a function *f* has an inverse then *f* is said to be **invertible**.

### 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 equivalent definition of an inverse function *g* of *f*, is to require that *g* o *f* be the identity function on the domain of *f*, and *f* o *g* be the identity function on the codomain of *f*, where "o" represents function composition.

### Existence

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

- (
*f*is onto) 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. - (
*f*is one-to-one) 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 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

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.

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

de:Umkehrfunktion fr:Application réciproque he:פונקציה הפיכה io:Simetra elemento pl:funkcja odwrotna uk:Обернена функція