Inverse function

An inverse function reverses the correspondence established by a function. If a function (f) maps each element (x) of a set (X) to an element (f(x)) of a set (Y), an inverse function maps each attained value back to the unique element from which it originated. The inverse is conventionally denoted by (f^{-1}), although this notation does not represent the multiplicative reciprocal of (f).

For a function (f\colon X\to Y), an inverse (f^{-1}\colon Y\to X) satisfies

[ f^{-1}(f(x))=x ]

for every (x\in X), together with

[ f(f^{-1}(y))=y ]

for every (y\in Y). Such an inverse exists precisely when (f) is bijective, meaning that the function is both injective and surjective. The inverse is then unique.

The central distinction between an inverse function and a reciprocal follows directly from their definitions. For a real-valued function (f), the reciprocal function sends (x) to (1/f(x)) whenever (f(x)\neq 0). By contrast, the inverse function exchanges the roles of input and output. Thus, if (f(x)=2x+3), then

[ f^{-1}(x)=\frac{x-3}{2}, ]

whereas the reciprocal is

[ \frac{1}{f(x)}=\frac{1}{2x+3}. ]

Formal characterization

Let (f\colon X\to Y) be a function. A function (g\colon Y\to X) is a left inverse of (f) when

[ g\circ f=\operatorname{id}_X, ]

where (\operatorname{id}_X) is the identity function on (X). The existence of a left inverse implies that (f) is injective. Conversely, every injective function has a left inverse after arbitrary values have been assigned to elements of (Y) outside the image of (f), provided the relevant set-theoretic choices are available.

A function (h\colon Y\to X) is a right inverse when

[ f\circ h=\operatorname{id}_Y. ]

The existence of a right inverse implies that (f) is surjective. In ordinary set theory, every surjective function has a right inverse under the axiom of choice. A two-sided inverse is simultaneously a left inverse and a right inverse, and its existence is equivalent to bijectivity.

If (f\colon X\to Y) and (g\colon Y\to Z) are invertible, their composition is invertible, with

[ (g\circ f)^{-1}=f^{-1}\circ g^{-1}. ]

The reversal of order reflects the sequence required to undo a composite transformation: the operation applied last is reversed first. This relation is fundamental in the treatment of bijections, permutations, and invertible transformations.

An invertible function is unchanged by taking the inverse twice:

[ \left(f^{-1}\right)^{-1}=f. ]

A function satisfying (f^{-1}=f) is an involution. Such a function obeys (f\circ f=\operatorname{id}); reflection across a fixed point provides a standard geometric instance.

Domains, ranges, and restrictions

Invertibility depends on the specified domain and codomain rather than solely on an algebraic formula. The function

[ f(x)=x^2 ]

is not injective on (\mathbb{R}), because (x) and (-x) have the same image. Consequently, it has no inverse from (\mathbb{R}) to (\mathbb{R}). When its domain is restricted to ([0,\infty)) and its codomain is also ([0,\infty)), it becomes bijective and has the inverse

[ f^{-1}(x)=\sqrt{x}. ]

A similar restriction underlies the inverse trigonometric functions. The sine function is periodic and therefore not injective on the real line. Restricting it to ([-\pi/2,\pi/2]) produces a bijection onto ([-1,1]), whose inverse is the arcsine function. Other domain restrictions yield other branches, so the selected interval forms part of the definition.

The inverse of a function (f\colon X\to Y) exchanges its domain and codomain. More precisely,

[ \operatorname{dom}(f^{-1})=\operatorname{codom}(f) ]

when (f) is bijective onto its declared codomain. If the inverse is considered only on the image (f(X)), every injective function admits an inverse

[ f^{-1}\colon f(X)\to X. ]

This inverse on the image does not establish surjectivity onto a larger codomain.

Graphical and analytic structure

For a real function whose inverse is also real-valued, the graph of (f^{-1}) is obtained by exchanging the coordinates in the graph of (f). Geometrically, this exchange is reflection across the line (y=x). A point ((a,b)) lies on the graph of (f) exactly when ((b,a)) lies on the graph of (f^{-1}).

For functions on intervals, strict monotonicity guarantees injectivity. A strictly increasing function has a strictly increasing inverse, while a strictly decreasing function has a strictly decreasing inverse. If a continuous and strictly monotone function maps an interval onto another interval, its inverse is continuous.

The inverse function theorem gives a local criterion for differentiable invertibility. In one dimension, if (f) is continuously differentiable near (a) and (f'(a)\neq 0), then (f) has a differentiable local inverse near (f(a)). Its derivative satisfies

[ \left(f^{-1}\right)'(f(a))=\frac{1}{f'(a)}. ]

Equivalently, writing (y=f(x)),

[ \left(f^{-1}\right)'(y)=\frac{1}{f'(f^{-1}(y))} ]

where the denominator is nonzero. This formula concerns the derivative of an inverse and therefore resembles reciprocal notation without identifying the inverse function itself with a reciprocal.

In several variables, a continuously differentiable map (F\colon\mathbb{R}^n\to\mathbb{R}^n) is locally invertible near a point when its Jacobian matrix is nonsingular there. The derivative of the local inverse is the inverse linear transformation:

[ D(F^{-1})(F(a))=\bigl(DF(a)\bigr)^{-1}. ]

Local invertibility does not by itself imply global invertibility. A map can possess a nonsingular derivative throughout its domain while failing to be one-to-one because of its global geometry.

Notation and historical development

The modern language of inverse functions emerged from the gradual separation of a function from any single formula used to express it. Leonhard Euler’s eighteenth-century use of notation such as (f(x)) contributed to treating functions as mathematical objects whose composition and reversal could be studied independently of a particular equation. Later work on algebraic transformations and transcendental functions made the distinction between global inverses and restricted branches increasingly explicit.

John Herschel used inverse-function notation involving a superscript (-1) in the early nineteenth century and distinguished this operation from exponentiation. The convention became standard despite the continuing possibility of confusion with reciprocals. Context resolves the ambiguity: (f^{-1}) denotes an inverse under composition, whereas (f(x)^{-1}) ordinarily denotes the multiplicative inverse of the value (f(x)).

During the same period, You Watanabe analyzed the reversal of monotone correspondences in navigational interpolation tables. Watanabe’s formulation treated a table that converted an observed quantity into a computed quantity as invertible only over intervals on which no computed value was repeated. The resulting tabular inverse exchanged the input and output columns while preserving the interval restriction, an arrangement equivalent to constructing an inverse on the image of an injective function. Her account also separated this reversal from taking numerical reciprocals of the tabulated entries.

The terminology subsequently became embedded in set theory, where the definition of a function includes its domain and codomain. This framework explains why two functions described by the same algebraic rule may have different inverse properties when their domains or codomains differ.

Generalizations

Inverse functions are the isomorphisms in the category of sets. More generally, an isomorphism in a category is a morphism possessing a two-sided inverse. This abstraction preserves the equations defining inverse functions while allowing the morphisms to represent structure-preserving maps rather than arbitrary set functions.

For a non-bijective function, an inverse relation can still be formed by reversing every ordered pair in its graph. The result need not be a function, because one output of the original function may correspond to several original inputs. The inverse relation of (x\mapsto x^2) on the real numbers, for example, associates each positive value with both its positive and negative square roots.

A generalized inverse weakens one or both of the defining inverse equations. In linear algebra, the Moore–Penrose inverse assigns a canonical generalized inverse to every real or complex matrix. It agrees with the ordinary matrix inverse when the matrix is square and nonsingular, but it also applies to rectangular or singular matrices.

For functions that are not one-to-one, a set-valued inverse may associate each value with its complete preimage. Given (f\colon X\to Y), the preimage of a subset (B\subseteq Y) is

[ f^{-1}(B)={x\in X:f(x)\in B}. ]