- Inverse function
In

mathematics , if ƒ is a function from "A" to "B" then an**inverse function**for ƒ is a function in the opposite direction, from "B" to "A", with the property that a round trip (a composition) from "A" to "B" to "A" (or from "B" to "A" to "B") returns each element of the initial set to itself. Thus, if an input "x" into the function ƒ produces an output "y", then inputting "y" into the inverse function ƒ^{–1}produces the output "x". Not every function has an inverse; those that do are called**invertible**.For example, let ƒ be the function that converts a temperature in degrees

Celsius to a temperature in degreesFahrenheit ::$f(C)\; =\; frac95\; C\; +\; 32\; ;\; ,!$then its inverse function converts degrees Fahrenheit to degrees Celsius::$f^\{-1\}(F)\; =\; frac59\; (F\; -\; 32)\; .\; ,!$Or, suppose ƒ assigns each child in a family of three the year of its birth. An inverse function would tell us which child was born in a given year. However, if the family has twins (or triplets) then we cannot know which to name for their common birth year. As well, if we are given a year in which no child was born then we cannot name a child. 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,:$egin\{align\}\; f(\; ext\{Alan\})=2005\; ,\; quad\; f(\; ext\{Brad\})=2007\; ,\; quad\; f(\; ext\{Cary\})=2001\; \backslash \; f^\{-1\}(2001)=\; ext\{Cary\}\; ,\; quad\; f^\{-1\}(2005)=\; ext\{Alan\}\; ,\; quad\; f^\{-1\}(2007)=\; ext\{Brad\}end\{align\}$

**Definitions**Let ƒ be a function whose domain is the set "X", and whose range is the set "Y". Then, if it exists, the

**inverse**of ƒ is the function ƒ^{–1}with domain "Y" and range "X", defined by the following rule::$ext\{If\; \}f(x)\; =\; y\; ext\{,\; then\; \}f^\{-1\}(y)\; =\; x\; ext\{.\},!$Stated otherwise, a function is invertible if and only if its

inverse relation is a function, in which case the inverse relation is the inverse function: the inverse relation is the relation obtained by switching "x" and "y" everywhere.Thus, an inverse function uniquely identifies the input "x" of another function based only on its output "y", for all "y" ∈ "Y". A function is invertible if and only if this rule defines a function. Not all functions have an inverse. For this rule to be applicable, each element "y" ∈ "Y" must correspond to exactly one element "x" ∈ "X". This is generally stated as two conditions:

* Every $y\; in\; Y$ corresponds to**no more than one**$x\; in\; X$; a function ƒ with this property is called one-to-one, or information-preserving, or an injection.

* Every $y\; in\; Y$ corresponds to**at least one**$x\; in\; X$; a function ƒ with this property is called onto, or a surjection.In elementary mathematics, the domain is often assumed to be the real numbers, if not otherwise specified, and the range is assumed to be the image.

Most functions encountered in elementary calculus do not have an inverse. [

*Smith, William K. "Inverse Functions", MacMillan, 1966 (p. 60).*]**Example: square root**The function ƒ("x") = "y" = "x"

^{2}may or may not be invertible, depending on the domain and codomain.If the domain is the real numbers, then each element in "Y" would correspond to two different elements in "X" (±"x"), and therefore ƒ would not be invertible. More precisely, the square of "x" is not invertible because it is impossible to deduce from its output the sign of its input. Such a function is called non-injective or information-losing. Notice that neither the

square root nor theprincipal square root function is the inverse of "x"^{2}because the first is not single-valued, and the second returns -"x" when "x" is negative.If the domain and codomain are both the non-negative numbers, then it is invertible, by the

principal square root .If the domain is the non-negative numbers, but the codomain is all reals, then again, it is not invertible, because negative numbers are not squares of a real number.

**Inverses in higher mathematics**The definition given above is commonly adopted in

calculus . In higher mathematics, the notation:$fcolon\; X\; o\; Y\; ,!$means "ƒ is a function mapping elements of a set "X" to elements of a set "Y". The source, "X", is called the domain of ƒ, and the target, "Y", is called thecodomain . The codomain contains the range of ƒ as asubset , and is considered part of the definition of ƒ.When using codomains, the inverse of a function nowrap| ƒ: "X" → "Y" is required to have domain "Y" and codomain "X". For the inverse to be defined on all of "Y", every element of "Y" must lie in the range of the function ƒ. A function with this property is called onto or a surjection. Thus, a function with a codomain is invertible

if and only if it is both one-to-one and onto. Such a function is called a one-to-one correspondence or abijection , and has the property that every element nowrap| "y" ∈ "Y" corresponds to exactly one element nowrap| "x" ∈ "X".**Inverses and composition**If ƒ is an invertible function with domain "X" and range "Y", then

:$f^\{-1\}left(\; ,\; f(x)\; ,\; ight)\; =\; x\; ext\{,\; for\; every\; \}x\; in\; X\; ext\{.\}$

This statement is equivalent to the first of the above-given definitions of the inverse, and it becomes equivalent to the second definition if Y coincides with the codomain of ƒ. Using the

composition of functions we can rewrite this statement as follows::$f^\{-1\}\; circ\; f\; =\; mathrm\{id\}\_X\; ext\{,\}$

where id

_{"X"}is theidentity function on the set "X". Incategory theory , this statement is used as the definition of an inversemorphism .If we think of composition as a kind of multiplication of functions, this identity says that the inverse of a function is analogous to a

multiplicative inverse . This explains the origin of the notation ƒ^{–1}.**Note on notation**The superscript notation for inverses can sometimes be confused with other uses of superscripts, especially when dealing with trigonometric and hyperbolic functions.

It is important to realize that ƒ

^{–1}(x) is not the same as ƒ(x)^{–1}. In ƒ^{−1}("x"), the superscript "−1" is not anexponent . A similar notation is used indynamical system s foriterated function s. For example, ƒ^{2}denotes two iterations of the function ƒ; if nowrap|1= ƒ("x") = "x" + 1, then nowrap|1=ƒ^{2}("x") = ("x" + 1) + 1, or "x" + 2. In symbols:: $f^2(x)\; =\; f(f(x))\; =\; (f\; circ\; f)(x).$In calculus, ƒ

^{("n")}, with parentheses, denotes the "n"thderivative of a function ƒ. For instance:: $f^\{(2)\}(x)\; =\; frac\{d^\{2\{dx^\{2f(x).$In

trigonometry , for historical reasons, sin^{2}("x") usually "does" mean the square of sin("x")::$sin^2\; x\; =\; (sin\; x)^2.\; ,!$

However, the expression sin

^{-1}("x") "does not" represent the multiplicative inverse to sin("x")::$sin^\{-1\}\; x\; eq\; (sin\; x)^\{-1\}.\; ,!$

It denotes the inverse function for sin("x") (actually a partial inverse; see below). To avoid confusion, an

inverse trigonometric function is often indicated by the prefix "arc". For instance the inverse sine is typically called thearcsine ::$sin^\{-1\}\; x\; =\; arcsin\; x\; =\; mathrm\{asin\},\; x.\; ,!$

The function nowrap| (sin "x")

^{–1}is the multiplicative inverse to the sine, and is called thecosecant . It is usually denoted csc "x"::$(sin\; x)^\{-1\}\; =\; frac\{1\}\{sin\; x\}\; =\; csc\; x\; .\; ,!$**Properties****Uniqueness**If an inverse function exists for a given function ƒ, it is unique: it must be the

inverse relation .**ymmetry**There is a symmetry between a function and its inverse. Specifically, if the inverse of ƒ is ƒ

^{–1}, then the inverse of ƒ^{–1}is the original function ƒ. In symbols::$egin\{align\}\; ext\{If\; \}\; f^\{-1\}\; circ\; f\; =\; mathrm\{id\}\_X\; ext\{,\}\; \backslash \; ext\{then\; \}\; f\; circ\; f^\{-1\}\; =\; mathrm\{id\}\_Y\; ext\{.\}end\{align\}$

This follows because invertion of relations is an involution: if you repeat it, you get back to where you started.

This statement is an obvious consequence of the above-explained deduction that, for ƒ to be invertible, it must be injective (first definition of the inverse) or bijective (second definition). The property of symmetry can be concisely expressed by the following formula:

:$left(f^\{-1\}\; ight)^\{-1\}\; =\; f\; .\; ,!$

**Inverse of a composition**The inverse of a composition of functions is given by the formula:$(f\; circ\; g)^\{-1\}\; =\; g^\{-1\}\; circ\; f^\{-1\}$Notice that the order of ƒ and "g" have been reversed; to undo "g" followed by ƒ, we must first undo ƒ and then undo "g".

For example, let nowrap|1= ƒ("x") = "x" + 5, and let nowrap|1= "g"("x") = 3"x". Then the composition nowrap| ƒ o "g" is the function that first multiplies by three and then adds five::$(f\; circ\; g)(x)\; =\; 3x\; +\; 5$To reverse this process, we must first subtract five, and then divide by three::$(f\; circ\; g)^\{-1\}(y)\; =\; frac13(y\; -\; 5)$This is the composition nowrap| ("g"

^{–1}o ƒ^{–1}) ("y").**elf-inverses**If "X" is a set, then the

identity function on "X" is its own inverse::$mathrm\{id\}\_X^\{-1\}\; =\; mathrm\{id\}\_X$

More generally, a function nowrap| ƒ: "X" → "X" is equal to its own inverse if and only if the composition nowrap| ƒ o ƒ is equal to id

_{"x"}. Such a function is called aninvolution .**Inverses in calculus**Single-variable

calculus is primarily concerned with functions that mapreal number s to real numbers. Such functions are often defined throughformula s, such as::$f(x)\; =\; (2x\; +\; 8)^3\; .\; ,!$A function ƒ from the real numbers to the real numbers possesses an inverse as long as it is one-to-one, i.e. as long as the graph of the function passes thehorizontal line test .The following table shows several standard functions and their inverses::

**Formula for the inverse**One approach to finding a formula for ƒ

^{–1}, if it exists, is to solve the equation nowrap|1= "y" = ƒ("x") for "x". For example, if ƒ is the function:$f(x)\; =\; (2x\; +\; 8)^3\; ,!$

then we must solve the equation nowrap|1= "y" = (2"x" + 8)

^{3}for "x"::$egin\{align\}\; y\; =\; (2x+8)^3\; \backslash \; sqrt\; [3]\; \{y\}\; =\; 2x\; +\; 8\; \backslash sqrt\; [3]\; \{y\}\; -\; 8\; =\; 2x\; \backslash dfrac\{sqrt\; [3]\; \{y\}\; -\; 8\}\{2\}\; =\; x\; .end\{align\}$

Thus the inverse function ƒ

^{–1}is given by the formula:$f^\{-1\}(y)\; =\; dfrac\{sqrt\; [3]\; \{y\}\; -\; 8\}\{2\}\; .\; ,!$

Sometimes the inverse of a function cannot be expressed by a formula. For example, if ƒ is the function

:$f(x)\; =\; x\; +\; sin\; x\; ,\; ,!$

then ƒ is one-to-one, and therefore possesses an inverse function ƒ

^{–1}. There is no simple formula for this inverse, since the equation nowrap|1= "y" = "x" + sin "x" cannot be solved algebraically for "x".**Graph of the inverse**If ƒ and ƒ

^{–1}are inverses, then the graph of the function:$y\; =\; f^\{-1\}(x),!$

is the same as the graph of the equation

:$x\; =\; f(y)\; .\; ,!$

This is identical to the equation nowrap|1= "y" = ƒ("x") that defines the graph of ƒ, except that the roles of "x" and "y" have been reversed. Thus the graph of ƒ

^{–1}can be obtained from the graph of ƒ by switching the positions of the "x" and "y" axes. This is equivalent to reflecting the graph across the linenowrap|1= "y" = "x".**Inverses and derivatives**A

continuous function ƒ is one-to-one (and hence invertible) 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 nowrap|1= ƒ′("x") = 3"x"^{2}+ 1 is always positive.If the function ƒ is

differentiable , then the inverse ƒ^{–1}will be differentiable as long as nowrap| ƒ′("x") ≠ 0. The derivative of the inverse is given by theinverse function theorem ::$frac\{d\}\{dy\}left\; [\; f^\{-1\}(y)\; ight]\; =\; frac\{1\}\{f\text{'}left(f^\{-1\}(y)\; ight)\}\; .$If we set nowrap|1= "x" = ƒ^{–1}("y"), then the formula above can be written:$frac\{dx\}\{dy\}\; =\; frac\{1\}\{dy\; /\; dx\}\; .$This result follows from thechain rule (see the article oninverse functions and differentiation ).The inverse function theorem can be generalized to functions of several variables. Specifically, a differentiable function nowrap| ƒ:

**R**^{"n"}→**R**^{"n"}is invertible in a neighborhood of a point "p" as long as theJacobian matrix of ƒ at "p" is invertible. In this case, the Jacobian of ƒ^{–1}at ƒ("p") is thematrix inverse of the Jacobian of ƒ at "p".**Generalizations****Partial inverses**Even if a function ƒ is not one-to-one, it may be possible to define a

**partial inverse**of ƒ by restricting the domain. For example, the function:$f(x)\; =\; x^2,!$

is not one-to-one, since nowrap|1= "x"

^{2}= (–"x")^{2}. However, the function becomes one-to-one if we restrict to the domain nowrap| "x" ≥ 0, in which case:$f^\{-1\}(y)\; =\; sqrt\{y\}\; .$

(If we instead restrict to the domain nowrap| "x" ≤ 0, then the inverse is the negative of the square root of "x".) Alternatively, there is no need to restrict the domain if we are content with the inverse being a

multivalued function ::$f^\{-1\}(y)\; =\; pmsqrt\{y\}\; .$

Sometimes this multivalued inverse is called the

**full inverse**of ƒ, and the portions (such as √"x" and −√"x") are called**branches**. The most important branch of a multivalued function (e.g. the positive square root) is called the**principal branch**, and its value at "y" is called the**principal value**of ƒ^{–1}("y").For a continuous function on the real line, one branch is required between each pair of local extrema. For example, the inverse of a

cubic function with a local maximum and a local minimum has three branches (see the picture to the right).These considerations are particularly important for defining the inverses of

trigonometric functions . For example, thesine function is not one-to-one, since:$sin(x\; +\; 2pi)\; =\; sin(x),!$

for every real "x" (and more generally nowrap|1= sin("x" + 2π"n") = sin("x") for every

integer "n"). However, the sine is one-to-one on the intervalnowrap| [–^{π}⁄_{2},^{π}⁄_{2}] , and the corresponding partial inverse is called thearcsine . This is considered the principal branch of the inverse sine, so the principal value of the inverse sine is always between –^{π}⁄_{2}and^{π}⁄_{2}. The following table describes the principal branch of each inverse trigonometric function::**Left and right inverses**If ƒ: "X" → "Y", a

**left inverse**for ƒ (or**retraction**of ƒ) is a function nowrap| "g": "Y" → "X" such that:$g\; circ\; f\; =\; mathrm\{id\}\_X\; .\; ,!$

That is, the function "g" satisfies the rule

:$ext\{If\; \}f(x)\; =\; y\; ext\{,\; then\; \}g(y)\; =\; x\; .\; ,!$

Thus, "g" must equal the inverse of ƒ on the range of ƒ, but may take any values for elements of "Y" not in the range. A function ƒ has a left inverse if and only if it is injective.

A

**right inverse**for ƒ (or**section**of ƒ) is a function nowrap| "h": "Y" → "X" such that:$f\; circ\; h\; =\; mathrm\{id\}\_Y\; .\; ,!$

That is, the function "h" satisfies the rule

:$ext\{If\; \}h(y)\; =\; x\; ext\{,\; then\; \}f(x)\; =\; y\; .\; ,!$

Thus, "h"("y") may be any of the elements of "x" that map to "y" under ƒ. A function ƒ has a right inverse if and only if it is surjective (though constructing such an inverse in general requires the

axiom of choice ).An inverse which is both a left and right inverse must be unique; otherwise not. Likewise, if "g" is a left inverse for ƒ then ƒ may not be a right inverse for "g"; and if ƒ is a right inverse for "g" then "g" is not necessarily a left inverse for ƒ.

=PreIf ƒ: "X" → "Y" is any function (not necessarily invertible), the

**preimage**(or**inverse image**) of an element nowrap| "y" ∈ "Y" is the set of all elements of "X" that map to "y"::$f^\{-1\}(y)\; =\; left\{\; xin\; X\; :\; f(x)\; =\; y\; ight\}\; .\; ,!$

The preimage of "y" can be thought of as the image of "y" under the (multivalued) full inverse of the function "f".

Similarly, if "S" is any

subset of "Y", the preimage of "S" is the set of all elements of "X" that map to "S"::$f^\{-1\}(S)\; =\; left\{\; xin\; X\; :\; f(x)\; in\; S\; ight\}\; .\; ,!$

The preimage of a single element nowrap| "y" ∈ "Y" is sometimes called the

**fiber**of "y". When "Y" is the set of real numbers, it is common to refer to ƒ^{–1}("y") as a.level set **ee also***

Inverse trigonometric function

*Logarithm

*Inverse function theorem

*Inverse functions and differentiation

*Inverse relation

*Inverse element **References****Bibliography*** Citation

last = Stewart

first = James

date = 2002

title = Calculus

publisher = Brooks Cole

edition = 5th

isbn = 978-0534393397

*Wikimedia Foundation.
2010.*