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There is a symmetry between a function and its inverse. Specifically, if f is an invertible function with domain X and codomain Y, then its inverse f −1 has domain Y and image X, and the inverse of f −1 is the original function f. In symbols, for functions f:X → Y and f −1:Y → X, [13]
In calculus, the inverse function rule is a formula that expresses the derivative of the inverse of a bijective and differentiable function f in terms of the derivative of f. More precisely, if the inverse of f {\displaystyle f} is denoted as f − 1 {\displaystyle f^{-1}} , where f − 1 ( y ) = x {\displaystyle f^{-1}(y)=x} if and only if f ...
For functions of a single variable, the theorem states that if is a continuously differentiable function with nonzero derivative at the point ; then is injective (or bijective onto the image) in a neighborhood of , the inverse is continuously differentiable near = (), and the derivative of the inverse function at is the reciprocal of the derivative of at : ′ = ′ = ′ (()).
The inverse function of a composition (assumed invertible) has the property that (f ∘ g) −1 = g −1 ∘ f −1. [4] Derivatives of compositions involving differentiable functions can be found using the chain rule. Higher derivatives of such functions are given by Faà di Bruno's formula. [3]
Since and the inverse function : are continuous, they have antiderivatives by the fundamental theorem of calculus. Laisant proved that if F {\displaystyle F} is an antiderivative of f {\displaystyle f} , then the antiderivatives of f − 1 {\displaystyle f^{-1}} are:
Functions that have inverse functions are said to be invertible. A function is invertible if and only if it is a bijection. Stated in concise mathematical notation, a function f: X → Y is bijective if and only if it satisfies the condition for every y in Y there is a unique x in X with y = f(x).
For a function to have an inverse, it must be one-to-one.If a function is not one-to-one, it may be possible to define a partial inverse of by restricting the domain. For example, the function = defined on the whole of is not one-to-one since = for any .
An involution is a function f : X → X that, when applied twice, brings one back to the starting point. In mathematics, an involution, involutory function, or self-inverse function [1] is a function f that is its own inverse, f(f(x)) = x. for all x in the domain of f. [2] Equivalently, applying f twice produces the original value.