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In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, the inverse function of a
function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
(also called the inverse of ) is a
function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
that undoes the operation of . The inverse of exists if and only if is
bijective In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

bijective
, and if it exists, is denoted by f^. For a function f\colon X\to Y, its inverse f^\colon Y\to X admits an explicit description: it sends each element y\in Y to the unique element x\in X such that . As an example, consider the
real-valued In mathematics, value may refer to several, strongly related notions. In general, a mathematical value may be any definite mathematical object. In elementary mathematics, this is most often a number – for example, a real number such as or an in ...
function of a real variable given by . One can think of as the function which multiplies its input by 5 then subtracts 7 from the result. To undo this, one adds 7 to the input, then divides the result by 5. Therefore, the inverse of is the function f^\colon \R\to\R defined by f^(y) = (y+7)/5.


Definitions

Let be a function whose
domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined **Domain of definition of a partial function **Natural domain of a partial function **Domain of holomorphy of a function *Doma ...
is the set , and whose
codomain In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

codomain
is the set . Then is ''invertible'' if there exists a function from to such that g(f(x))=x for all x\in X and f(g(y))=y for all y\in Y. If is invertible, then there is exactly one function satisfying this property. The function is called the inverse of , and is usually denoted as , a notation introduced by
John Frederick William Herschel Sir John Frederick William Herschel, 1st Baronet (; 7 March 1792 – 11 May 1871) was an English polymath A polymath ( el, πολυμαθής, ', "having learned much"; Latin Latin (, or , ) is a classical language belonging to th ...
in 1813. The function is invertible if and only if it is bijective. This is because the condition g(f(x))=x for all x\in X implies that is
injective In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

injective
, and the condition f(g(y))=y for all y\in Y implies that is
surjective In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

surjective
. The inverse function to can be explicitly described as the function :f^(y)=(\textx\in X\textf(x)=y).


Inverses and composition

Recall that if is an invertible function with domain and codomain , then : f^\left(f(x)\right) = x, for every x \in X and f\left(f^(y)\right) = y for every y \in Y . Using the
composition of functions In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
, we can rewrite this statement as follows: : f^ \circ f = \operatorname_X and f \circ f^ = \operatorname_Y, where is the
identity function Graph of the identity function on the real numbers In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function (mathematics), function that always returns the same value that was ...

identity function
on the set ; that is, the function that leaves its argument unchanged. In
category theory Category theory formalizes mathematical structure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and ...
, this statement is used as the definition of an inverse
morphism In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...

morphism
. Considering function composition helps to understand the notation . Repeatedly composing a function with itself is called
iteration Iteration is the repetition of a process in order to generate a (possibly unbounded) sequence of outcomes. Each repetition of the process is a single iteration, and the outcome of each iteration is then the starting point of the next iteration. ...
. If is applied times, starting with the value , then this is written as ; so , etc. Since , composing and yields , "undoing" the effect of one application of .


Notation

While the notation might be misunderstood, certainly denotes the
multiplicative inverse Image:Hyperbola one over x.svg, thumbnail, 300px, alt=Graph showing the diagrammatic representation of limits approaching infinity, The reciprocal function: . For every ''x'' except 0, ''y'' represents its multiplicative inverse. The graph forms a r ...

multiplicative inverse
of and has nothing to do with the inverse function of . In keeping with the general notation, some English authors use expressions like to denote the inverse of the sine function applied to (actually a partial inverse; see below). Other authors feel that this may be confused with the notation for the multiplicative inverse of , which can be denoted as . To avoid any confusion, an
inverse trigonometric function In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
is often indicated by the prefix " arc" (for Latin ). For instance, the inverse of the sine function is typically called the
arcsine In mathematics, the inverse trigonometric functions (occasionally also called arcus functions, antitrigonometric functions or cyclometric functions) are the inverse functions of the trigonometric functions (with suitably restricted Domain of a func ...

arcsine
function, written as . Similarly, the inverse of a
hyperbolic function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

hyperbolic function
is indicated by the prefix " ar" (for Latin ). For instance, the inverse of the
hyperbolic sine In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
function is typically written as . Other inverse special functions are sometimes prefixed with the prefix "inv", if the ambiguity of the notation should be avoided.


Examples


Squaring and square root functions

The function given by is not injective because (-x)^2=x^2 for all x\in\R. Therefore, is not invertible. If the domain of the function is restricted to the nonnegative reals, that is, we take the function f\colon [0,\infty)\to [0,\infty);\ x\mapsto x^2 with the same ''rule'' as before, then the function is bijective and so, invertible. The inverse function here is called the ''(positive) square root function'' and is denoted by x\mapsto\sqrt x.


Standard inverse functions

The following table shows several standard functions and their inverses:


Formula for the inverse

Many functions given by algebraic formulas possess a formula for their inverse. This is because the inverse f^ of an invertible function f\colon\R\to\R has an explicit description as : f^(y)=(\textx\in \R\textf(x)=y). This allows one to easily determine inverses of many functions that are given by algebraic formulas. For example, if is the function : f(x) = (2x + 8)^3 then to determine f^(y) for a real number , one must find the unique real number such that . This equation can be solved: : \begin y & = (2x+8)^3 \\ \sqrt & = 2x + 8 \\ \sqrt - 8 & = 2x \\ \dfrac & = x . \end Thus the inverse function is given by the formula : f^(y) = \frac 2. Sometimes, the inverse of a function cannot be expressed by a closed-form formula. For example, if is the function : f(x) = x - \sin x , then is a bijection, and therefore possesses an inverse function . The formula for this inverse has an expression as an infinite sum: : f^(y) = \sum_^\infty \frac \lim_ \left( \frac \left( \frac \theta \right)^n \right).


Properties

Since a function is a special type of
binary relation Binary may refer to: Science and technology Mathematics * Binary number In mathematics and digital electronics Digital electronics is a field of electronics The field of electronics is a branch of physics and electrical engineeri ...
, many of the properties of an inverse function correspond to properties of
converse relation In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
s.


Uniqueness

If an inverse function exists for a given function , then it is unique. This follows since the inverse function must be the converse relation, which is completely determined by .


Symmetry

There is a symmetry between a function and its inverse. Specifically, if is an invertible function with domain and codomain , then its inverse has domain and image , and the inverse of is the original function . In symbols, for functions and , :f^\circ f = \operatorname_X and f \circ f^ = \operatorname_Y. This statement is a consequence of the implication that for to be invertible it must be bijective. The involutory nature of the inverse can be concisely expressed by :\left(f^\right)^ = f. The inverse of a composition of functions is given by :(g \circ f)^ = f^ \circ g^. Notice that the order of and have been reversed; to undo followed by , we must first undo , and then undo . For example, let and let . Then the composition is the function that first multiplies by three and then adds five, : (g \circ f)(x) = 3x + 5. To reverse this process, we must first subtract five, and then divide by three, : (g \circ f)^(x) = \tfrac13(x - 5). This is the composition .


Self-inverses

If is a set, then the
identity function Graph of the identity function on the real numbers In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function (mathematics), function that always returns the same value that was ...

identity function
on is its own inverse: : ^ = \operatorname_X. More generally, a function is equal to its own inverse, if and only if the composition is equal to . Such a function is called an
involution Involution may refer to: * Involute, a construction in the differential geometry of curves * ''Agricultural Involution: The Processes of Ecological Change in Indonesia'', a 1963 study of intensification of production through increased labour input ...
.


Graph of the inverse

If is invertible, then the graph of the function : y = f^(x) is the same as the graph of the equation : x = f(y) . This is identical to the equation that defines the graph of , except that the roles of and have been reversed. Thus the graph of can be obtained from the graph of by switching the positions of the and axes. This is equivalent to reflecting the graph across the line .


Inverses and derivatives

A
continuous function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
is invertible on its range (image) if and only if it is either strictly increasing or decreasing (with no local maxima or minima). For example, the function : f(x) = x^3 + x is invertible, since the
derivative In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

derivative
is always positive. If the function is
differentiable In calculus (a branch of mathematics), a differentiable function of one Real number, real variable is a function whose derivative exists at each point in its Domain of a function, domain. In other words, the Graph of a function, graph of a differen ...

differentiable
on an interval and for each , then the inverse is differentiable on . If , the derivative of the inverse is given by the
inverse function theorem In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
, : \left(f^\right)^\prime (y) = \frac. Using
Leibniz's notation In calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimal In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zero. ...
the formula above can be written as : \frac = \frac. This result follows from the
chain rule In calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of ...
(see the article on
inverse functions and differentiation Inverse or invert may refer to: Science and mathematics * Inverse (logic), a type of conditional sentence which is an immediate inference made from another conditional sentence * Additive inverse (negation), the inverse of a number that, when add ...
). The inverse function theorem can be generalized to functions of several variables. Specifically, a differentiable multivariable function is invertible in a neighborhood of a point as long as the
Jacobian matrix In vector calculus Vector calculus, or vector analysis, is concerned with differentiation Differentiation may refer to: Business * Differentiation (economics), the process of making a product different from other similar products * Produ ...
of at is invertible. In this case, the Jacobian of at is the
matrix inverseIn linear algebra, an ''n''-by-''n'' square matrix In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calc ...
of the Jacobian of at .


Real-world examples

* Let be the function that converts a temperature in degrees
Celsius The degree Celsius is a unit of temperature on the Celsius scale, a temperature scale Scale of temperature is a methodology of calibrating the physical quantity temperature in metrology. Empirical scales measure temperature in relation to conv ...

Celsius
to a temperature in degrees
Fahrenheit The Fahrenheit scale ( or ) is a temperature scale Scale of temperature is a methodology of calibrating the physical quantity temperature in metrology. Empirical scales measure temperature in relation to convenient and stable parameters, such a ...

Fahrenheit
, F = f(C) = \tfrac95 C + 32 ; then its inverse function converts degrees Fahrenheit to degrees Celsius, C = f^(F) = \tfrac59 (F - 32) , since \begin f^ (f(C)) = & f^\left( \tfrac95 C + 32 \right) = \tfrac59 \left( (\tfrac95 C + 32 ) - 32 \right) = C, \\ & \text C, \text \\f\left(f^(F)\right) = & f\left(\tfrac59 (F - 32)\right) = \tfrac95 \left(\tfrac59 (F - 32)\right) + 32 = F, \\ & \text F. \end * Suppose assigns each child in a family its birth year. An inverse function would output which child was born in a given year. However, if the family has children born in the same year (for instance, twins or triplets, etc.) then the output cannot be known when the input is the common birth year. As well, if a year is given in which no child was born then a child cannot be named. But if each child was born in a separate year, and if we restrict attention to the three years in which a child was born, then we do have an inverse function. For example, \begin f(\text)&=2005 , \quad & f(\text)&=2007 , \quad & f(\text)&=2001 \\ f^(2005)&=\text , \quad & f^(2007)&=\text , \quad & f^(2001)&=\text \end * Let be the function that leads to an percentage rise of some quantity, and be the function producing an percentage fall. Applied to $100 with = 10%, we find that applying the first function followed by the second does not restore the original value of $100, demonstrating the fact that, despite appearances, these two functions are not inverses of each other. * The formula to calculate the pH of a solution is . In many cases we need to find the concentration of acid from a pH measurement. The inverse function is used.


Generalizations


Partial inverses

Even 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 : f(x) = x^2 is not one-to-one, since . However, the function becomes one-to-one if we restrict to the domain , in which case : f^(y) = \sqrt . (If we instead restrict to the domain , then the inverse is the negative of the square root of .) Alternatively, there is no need to restrict the domain if we are content with the inverse being a
multivalued function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

multivalued function
: : f^(y) = \pm\sqrt . Sometimes, this multivalued inverse is called the full inverse of , and the portions (such as and −) are called ''branches''. The most important branch of a multivalued function (e.g. the positive square root) is called the ''
principal branch In mathematics, a principal branch is a function which selects one branch point, branch ("slice") of a multi-valued function. Most often, this applies to functions defined on the complex plane. Examples Trigonometric inverses Principal bran ...
'', and its value at is called the ''principal value'' of . For a continuous function on the real line, one branch is required between each pair of
local extrema In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function (mathematics), function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, ei ...

local extrema
. For example, the inverse of a
cubic function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

cubic function
with a local maximum and a local minimum has three branches (see the adjacent picture). These considerations are particularly important for defining the inverses of
trigonometric functions In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

trigonometric functions
. For example, the
sine function In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side that is oppo ...

sine function
is not one-to-one, since : \sin(x + 2\pi) = \sin(x) for every real (and more generally for every
integer An integer (from the Latin Latin (, or , ) is a classical language A classical language is a language A language is a structured system of communication Communication (from Latin ''communicare'', meaning "to share" or "to ...
). However, the sine is one-to-one on the interval , and the corresponding partial inverse is called the
arcsine In mathematics, the inverse trigonometric functions (occasionally also called arcus functions, antitrigonometric functions or cyclometric functions) are the inverse functions of the trigonometric functions (with suitably restricted Domain of a func ...

arcsine
. This is considered the principal branch of the inverse sine, so the principal value of the inverse sine is always between − and . The following table describes the principal branch of each inverse trigonometric function: :


Left and right inverses

Left and right inverses are not necessarily the same. If is a left inverse for , then may or may not be a right inverse for ; and if is a right inverse for , then is not necessarily a left inverse for . For example, let denote the squaring map, such that for all in , and let denote the square root map, such that for all . Then for all in ; that is, is a right inverse to . However, is not a left inverse to , since, e.g., .


Left inverses

If , a left inverse for (or ''
retraction In academic publishing Academic publishing is the subfield of publishing which distributes academic research and scholarship. Most academic work is published in academic journal articles, books or thesis' form. The part of academic written o ...
'' of ) is a function such that composing with from the left gives the identity function: g \circ f = \operatorname_X . That is, the function satisfies the rule : If f(x) = y, then g(y) = x . Thus, must equal the inverse of on the image of , but may take any values for elements of not in the image. A function is injective if and only if it has a left inverse or is the empty function. : If is the left inverse of , then is injective. If , then g(f(x)) = g(f(y)) = x = y. : If is injective, either is the empty function () or has a left inverse (, which can be constructed as follows: for all , if is in the image of (there exists such that ), let ( is unique because is injective); otherwise, let be an arbitrary element of . For all , is in the image of , so by above, so is a left inverse of . In classical mathematics, every injective function with a nonempty domain necessarily has a left inverse; however, this may fail in
constructive mathematics In the philosophy of mathematics, constructivism asserts that it is necessary to find (or "construct") a mathematical object to prove that it exists. In classical mathematics, one can prove the existence of a mathematical object without "finding ...
. For instance, a left inverse of the inclusion of the two-element set in the reals violates
indecomposability In constructive mathematics, indecomposability or indivisibility (german: Unzerlegbarkeit, from the adjective ''unzerlegbar'') is the principle that the Continuum (set theory), continuum cannot be partition of a set, partitioned into two nonempty ...
by giving a
retraction In academic publishing Academic publishing is the subfield of publishing which distributes academic research and scholarship. Most academic work is published in academic journal articles, books or thesis' form. The part of academic written o ...
of the real line to the set .


Right inverses

A right inverse for (or ''
section Section, Sectioning or Sectioned may refer to: Arts, entertainment and media * Section (music) AABA form) in the key of C. ">Thirty-two-bar_form.html" ;"title="Rhythm changes bridge (B section of an Thirty-two-bar form">AABA form) in the key ...
'' of ) is a function such that : f \circ h = \operatorname_Y . That is, the function satisfies the rule : If \displaystyle h(y) = x, then \displaystyle f(x) = y . Thus, may be any of the elements of that map to under . A function has a right inverse if and only if it is surjective (though constructing such an inverse in general requires the
axiom of choice In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product#Infinite Cartesian products, Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the a ...

axiom of choice
). : If is the right inverse of , then is surjective. For all y \in Y, there is x = h(y) such that f(x) = f(h(y)) = y. : If is surjective, has a right inverse , which can be constructed as follows: for all y \in Y, there is at least one x \in X such that f(x) = y (because is surjective), so we choose one to be the value of .


Two-sided inverses

An inverse that is both a left and right inverse (a two-sided inverse), if it exists, must be unique. In fact, if a function has a left inverse and a right inverse, they are both the same two-sided inverse, so it can be called the inverse. : If g is a left inverse and h a right inverse of f, for all y \in Y, g(y) = g(f(h(y)) = h(y). A function has a two-sided inverse if and only if it is bijective. : A bijective function is injective, so it has a left inverse (if is the empty function, f \colon \varnothing \to \varnothing is its own left inverse). is surjective, so it has a right inverse. By the above, the left and right inverse are the same. : If has a two-sided inverse , then is a left inverse and right inverse of , so is injective and surjective.


Preimages

If is any function (not necessarily invertible), the preimage (or inverse image) of an element is defined to be the set of all elements of that map to : : f^(\) = \left\ . The preimage of can be thought of as the
image An image (from la, imago) is an artifact that depicts visual perception Visual perception is the ability to interpret the surrounding environment (biophysical), environment through photopic vision (daytime vision), color vision, sco ...
of under the (multivalued) full inverse of the function . Similarly, if is any
subset In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

subset
of , the preimage of , denoted f^(S) , is the set of all elements of that map to : : f^(S) = \left\ . For example, take the function . This function is not invertible as it is not bijective, but preimages may be defined for subsets of the codomain, e.g. : f^(\left\) = \left\. The preimage of a single element – a
singleton set In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
– is sometimes called the ''
fiber Fiber or fibre (from la, fibra, links=no) is a natural Nature, in the broadest sense, is the natural, physical, material world or universe The universe ( la, universus) is all of space and time and their contents, including ...
'' of . When is the set of real numbers, it is common to refer to as a ''
level set In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...
''.


See also

*
Lagrange inversion theorem In mathematical analysis, the Lagrange inversion theorem, also known as the Lagrange–Bürmann formula, gives the Taylor series In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (numbe ...
, gives the Taylor series expansion of the inverse function of an analytic function *
Integral of inverse functions In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
*
Inverse Fourier transformIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
*
Reversible computing Reversible computing is any model of computation A model is an informative representation of an object, person or system. The term originally denoted the plan A plan is typically any diagram or list of steps with details of timing and resources ...


Notes


References


Bibliography

* * * * * * *


Further reading

* * * *


External links

* {{springer, title=Inverse function, id=p/i052360
Basic concepts in set theory{{Commons This category is for the foundational concepts of naive set theory, in terms of which contemporary mathematics is typically expressed. Mathematical concepts ...
Unary operations