Inverse function

An inverse function is a concept of mathematics. A function will calculate some output [math]\displaystyle{ y }[/math], given some input [math]\displaystyle{ x }[/math]. This is usually written [math]\displaystyle{ f(x) = y }[/math]. The inverse function does the reverse. Let's say [math]\displaystyle{ g() }[/math] is the inverse function of [math]\displaystyle{ f() }[/math], then [math]\displaystyle{ g(y) = x }[/math]. Or otherwise put, [math]\displaystyle{ f(g(x)) = x }[/math]. An inverse function to [math]\displaystyle{ f(\ldots) }[/math] is usually called [math]\displaystyle{ f^{-1}(\ldots). }[/math] Do not confuse [math]\displaystyle{ f^{-1}(\ldots) }[/math] with [math]\displaystyle{ f(\ldots)^{-1} }[/math]: the first is a value of an inverse function, the second is reciprocal of a value of a normal function.

Examples

Let's take a function [math]\displaystyle{ f(x) = x^3 }[/math] over real [math]\displaystyle{ x }[/math]. Then, [math]\displaystyle{ f^{-1}(x) = \sqrt[3]{x}. }[/math]

Not all functions have inverse functions: for example, function [math]\displaystyle{ f(x) = |x| }[/math] has none (because [math]\displaystyle{ |-1| = 1 = |1| }[/math], and [math]\displaystyle{ f^{-1}(x) }[/math] should give both 1 and -1 when given 1)), but every binary relation has its own inverse relation.

Finding the inverse of a function can be very difficult to do.