# 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:

- .

*x*→ 3

*x*+ 2 is given, then its inverse function is

*x*→ (

*x*- 2) / 3. This is usually written as:

*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, 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*.

*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.

*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

*x*, and then exchanging

*y*and

*x*to get

*f*(

*x*) is analytic, the Lagrange inversion theorem may be used.