Inverse of a function.

The inverse of a function "reverses" or "undoes" what the function does.

Example: the cube root function is the inverse of the cubing function. Taking the cube root of a number that has been cubed "undoes" the cubing, leaving you back where you started. And vice versa, the cubing function is the inverse of the cube root function.
Simplest example? 2x doubles the argument. Its inverse is the function that halves its argument: ½ x. And vice versa, the halving function will be "undone" by the doubling function.
Simplester example? x+1 adds one to the argument. Its inverse subtracts one from its argument: x-1. Or, going the other way, first subtract 1, then add 1 leaves you where you started.

Having an inverse is a feature of some, not all, functions; see below for what determines whether a function has an inverse or not and for the usual suspects examples.

The inverse of a function ƒ is usually indicated as ƒ-1. (This is not an exponent and does not mean reciprocation.) Or othertimes by something boring like g.

Applying the inverse function to the output of the function (i.e. composing the function with its inverse) yields the original x value:
ƒ-1(ƒ(x)) = x    and    ƒ(ƒ-1(x)) = x
Another way to denote this is, using g for the inverse of ƒ:
g(ƒ(x))=x and ƒ(g(x))=x
Or, if ƒ(x)=y, then g(y)=x.
Another way to denote this is: (ƒ-1 ∘ ƒ)(x) = x and (ƒ ∘ ƒ-1)(x) = x
This happens with every x in the domain of ƒ.
This also means that a function composed with its inverse yields the identity function: ƒ-1 ∘ ƒ = id(x).

The range of ƒ is the domain of its inverse ƒ-1. And the domain of ƒ is the range of ƒ-1.

If a function has an inverse, it is unique, i.e. a function can have only one inverse. And the inverse of the inverse is this function, i.e. each is the inverse of the other, and no other.

To have an inverse, a function has to pass the horizontal line test: no horizontal line can touch the function's graph more than once. This means the function must be monotonically increasing or decreasing, i.e. always rising or always falling, which puts a big restriction on the number of functions that have an inverse, but see below for some colorful examples. This also means the function is one-to-one (1-1, or injective), i.e. each y value in its range is the function value of exactly one x value in its domain.
Example: ƒ(x)=x2 is not 1-1, doesn't pass the horizontal line test, is not montonically increasing or decreasing and so does not have an inverse.

Inverse functions are reflections of each other across (i.e. are symmetric about) the main diagonal y=x.
If a point (a,b) is on the graph of ƒ, then (b,a) is on the graph of ƒ-1.

Given the y=ƒ(x) formula for a function, with ƒ(x) an expression in x, to find its inverse, interchange (swap) the variables and then solve for y.


Cube root composed with cube:
Cube composed with cube root:
Both = x.

Every linear function has an inverse. The inverse of a linear function y=mx+b is y = (1/m)x - b/m, another linear function. Note the slopes are reciprocals of each other (and they have the same sign).
A linear function and its inverse intersect on the main diagonal y=x at the point with x and y coordinates of b/(1-m).

x2 does not have an inverse, but with a domain restricted to non-negative numbers it does.

x to an odd power and that oddth root of x are inverses: xodd and odd√x
x to an even power and that eventh root of x with domain ≥0 are inverses: xeven and even√x, x≥0

The exponential function and the natural logarithm function are inverses.

Each exponential function bx has logb x as its inverse.

A function can be its own inverse.
1/x is such an example. The reciprocal of the reciprocal of x is x.

ƒ(x)=x and ƒ(x)=-x are also self-inverses.

Sometimes it's not easy to find the inverse of a function and/or the inverse is complicated.
ƒ(x)=x3+x has inverse:

The trig functions on restricted domains give rise to the inverse trig functions: the arcsin, arccos, arctan functons. AKA sin-1, cos-1, tan-1 AKA asin, acos, atan