Английская Википедия:Inverse function

Шаблон:Functions In mathematics, the inverse function of a function Шаблон:Mvar (also called the inverse of Шаблон:Mvar) is a function that undoes the operation of Шаблон:Mvar. The inverse of Шаблон:Mvar exists if and only if Шаблон:Mvar is bijective, and if it exists, is denoted by <math>f^{-1} .</math>

For a function <math>f\colon X\to Y</math>, its inverse <math>f^{-1}\colon Y\to X</math> admits an explicit description: it sends each element <math>y\in Y</math> to the unique element <math>x\in X</math> such that Шаблон:Math.

As an example, consider the real-valued function of a real variable given by Шаблон:Math. One can think of Шаблон:Mvar 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 Шаблон:Mvar is the function <math>f^{-1}\colon \R\to\R</math> defined by <math>f^{-1}(y) = \frac{y+7}{5} .</math>

Definitions

Файл:Inverse Functions Domain and Range.png

If Шаблон:Mvar maps Шаблон:Mvar to Шаблон:Mvar, then Шаблон:Math maps Шаблон:Mvar back to Шаблон:Mvar.

Let Шаблон:Mvar be a function whose domain is the set Шаблон:Mvar, and whose codomain is the set Шаблон:Mvar. Then Шаблон:Mvar is invertible if there exists a function Шаблон:Mvar from Шаблон:Mvar to Шаблон:Mvar such that <math>g(f(x))=x</math> for all <math>x\in X</math> and <math>f(g(y))=y</math> for all <math>y\in Y</math>.^[1]

If Шаблон:Mvar is invertible, then there is exactly one function Шаблон:Mvar satisfying this property. The function Шаблон:Mvar is called the inverse of Шаблон:Mvar, and is usually denoted as Шаблон:Math, a notation introduced by John Frederick William Herschel in 1813.^[2]^[3]^[4]^[5]^[6]^{[nb 1]}

The function Шаблон:Mvar is invertible if and only if it is bijective. This is because the condition <math>g(f(x))=x</math> for all <math>x\in X</math> implies that Шаблон:Mvar is injective, and the condition <math>f(g(y))=y</math> for all <math>y\in Y</math> implies that Шаблон:Mvar is surjective.

The inverse function Шаблон:Math to Шаблон:Mvar can be explicitly described as the function

<math>f^{-1}(y)=(\text{the unique element }x\in X\text{ such that }f(x)=y)</math>.

Шаблон:AnchorInverses and composition

Шаблон:See also

Recall that if Шаблон:Mvar is an invertible function with domain Шаблон:Mvar and codomain Шаблон:Mvar, then

<math> f^{-1}\left(f(x)\right) = x</math>, for every <math>x \in X</math> and <math> f\left(f^{-1}(y)\right) = y</math> for every <math>y \in Y </math>.

Using the composition of functions, this statement can be rewritten to the following equations between functions:

<math> f^{-1} \circ f = \operatorname{id}_X</math> and <math>f \circ f^{-1} = \operatorname{id}_Y, </math>

where Шаблон:Math is the identity function on the set Шаблон:Mvar; that is, the function that leaves its argument unchanged. In category theory, this statement is used as the definition of an inverse morphism.

Considering function composition helps to understand the notation Шаблон:Math. Repeatedly composing a function Шаблон:Math with itself is called iteration. If Шаблон:Mvar is applied Шаблон:Mvar times, starting with the value Шаблон:Mvar, then this is written as Шаблон:Math; so Шаблон:Math, etc. Since Шаблон:Math, composing Шаблон:Math and Шаблон:Math yields Шаблон:Math, "undoing" the effect of one application of Шаблон:Mvar.

Notation

While the notation Шаблон:Math might be misunderstood,^[1] Шаблон:Math certainly denotes the multiplicative inverse of Шаблон:Math and has nothing to do with the inverse function of Шаблон:Mvar.^[6] The notation <math>f^{\langle -1\rangle}</math> might be used for the inverse function to avoid ambiguity with the multiplicative inverse.^[7]

In keeping with the general notation, some English authors use expressions like Шаблон:Math to denote the inverse of the sine function applied to Шаблон:Mvar (actually a partial inverse; see below).^[8]^[6] Other authors feel that this may be confused with the notation for the multiplicative inverse of Шаблон:Math, which can be denoted as Шаблон:Math.^[6] To avoid any confusion, an inverse trigonometric function is often indicated by the prefix "arc" (for Latin Шаблон:Lang).^[9]^[10] For instance, the inverse of the sine function is typically called the arcsine function, written as Шаблон:Math.^[9]^[10] Similarly, the inverse of a hyperbolic function is indicated by the prefix "ar" (for Latin Шаблон:Lang).^[10] For instance, the inverse of the hyperbolic sine function is typically written as Шаблон:Math.^[10] The expressions like Шаблон:Math can still be useful to distinguish the multivalued inverse from the partial inverse: <math>\sin^{-1}(x) = \{(-1)^n \arcsin(x) + \pi n : n \in \mathbb Z\}</math>. Other inverse special functions are sometimes prefixed with the prefix "inv", if the ambiguity of the Шаблон:Math notation should be avoided.^[11]^[10]

Examples

Squaring and square root functions

The function Шаблон:Math given by Шаблон:Math is not injective because <math>(-x)^2=x^2</math> for all <math>x\in\R</math>. Therefore, Шаблон:Mvar is not invertible.

If the domain of the function is restricted to the nonnegative reals, that is, we take the function <math>f\colon [0,\infty)\to [0,\infty);\ x\mapsto x^2</math> with the same rule as before, then the function is bijective and so, invertible.^[12] The inverse function here is called the (positive) square root function and is denoted by <math>x\mapsto\sqrt x</math>.

Standard inverse functions

The following table shows several standard functions and their inverses:

Inverse arithmetic functions
Function Шаблон:Math	Inverse Шаблон:Math	Notes
Шаблон:Math	Шаблон:Math
Шаблон:Math	Шаблон:Math
Шаблон:Math	Шаблон:Sfrac	Шаблон:Math
Шаблон:Sfrac (i.e. Шаблон:Math)	Шаблон:Sfrac (i.e. Шаблон:Math)	Шаблон:Math
Шаблон:Math	<math>\sqrt[p]y</math> (i.e. Шаблон:Math)	Шаблон:Math if Шаблон:Math is even; integer Шаблон:Math
Шаблон:Math	Шаблон:Math	Шаблон:Math and Шаблон:Math
Шаблон:Math	Шаблон:Math	Шаблон:Math and Шаблон:Math
trigonometric functions	inverse trigonometric functions	various restrictions (see table below)
hyperbolic functions	inverse hyperbolic functions	various restrictions

Formula for the inverse

Many functions given by algebraic formulas possess a formula for their inverse. This is because the inverse <math>f^{-1} </math> of an invertible function <math>f\colon\R\to\R</math> has an explicit description as

<math>f^{-1}(y)=(\text{the unique element }x\in \R\text{ such that }f(x)=y)</math>.

This allows one to easily determine inverses of many functions that are given by algebraic formulas. For example, if Шаблон:Mvar is the function

then to determine <math>f^{-1}(y) </math> for a real number Шаблон:Mvar, one must find the unique real number Шаблон:Mvar such that Шаблон:Math. This equation can be solved:

<math>\begin{align}

     y         & = (2x+8)^3 \\
 \sqrt[3]{y}   & = 2x + 8   \\

\sqrt[3]{y} - 8 & = 2x \\ \dfrac{\sqrt[3]{y} - 8}{2} & = x . \end{align}</math>

Thus the inverse function Шаблон:Math is given by the formula

Sometimes, the inverse of a function cannot be expressed by a closed-form formula. For example, if Шаблон:Mvar is the function

then Шаблон:Mvar is a bijection, and therefore possesses an inverse function Шаблон:Math. The formula for this inverse has an expression as an infinite sum:

<math> f^{-1}(y) =

\sum_{n=1}^\infty

\frac{y^{n/3}}{n!} \lim_{ \theta \to 0} \left(
\frac{\mathrm{d}^{\,n-1}}{\mathrm{d} \theta^{\,n-1}} \left(
\frac \theta { \sqrt[3]{ \theta - \sin( \theta )} } \right)^n

\right). </math>

Properties

Since a function is a special type of binary relation, many of the properties of an inverse function correspond to properties of converse relations.

Uniqueness

If an inverse function exists for a given function Шаблон:Mvar, then it is unique.^[13] This follows since the inverse function must be the converse relation, which is completely determined by Шаблон:Mvar.

Symmetry

There is a symmetry between a function and its inverse. Specifically, if Шаблон:Mvar is an invertible function with domain Шаблон:Mvar and codomain Шаблон:Mvar, then its inverse Шаблон:Math has domain Шаблон:Mvar and image Шаблон:Mvar, and the inverse of Шаблон:Math is the original function Шаблон:Mvar. In symbols, for functions Шаблон:Math and Шаблон:Math,^[13]

<math>f^{-1}\circ f = \operatorname{id}_X </math> and <math> f \circ f^{-1} = \operatorname{id}_Y.</math>

This statement is a consequence of the implication that for Шаблон:Mvar to be invertible it must be bijective. The involutory nature of the inverse can be concisely expressed by^[14]

<math>\left(f^{-1}\right)^{-1} = f.</math>

Файл:Composition of Inverses.png

The inverse of Шаблон:Math is Шаблон:Math.

The inverse of a composition of functions is given by^[15]

Notice that the order of Шаблон:Mvar and Шаблон:Mvar have been reversed; to undo Шаблон:Mvar followed by Шаблон:Mvar, we must first undo Шаблон:Mvar, and then undo Шаблон:Mvar.

For example, let Шаблон:Math and let Шаблон:Math. Then the composition Шаблон:Math is the function that first multiplies by three and then adds five,

To reverse this process, we must first subtract five, and then divide by three,

<math>(g \circ f)^{-1}(x) = \tfrac13(x - 5).</math>

This is the composition Шаблон:Math.

Self-inverses

If Шаблон:Mvar is a set, then the identity function on Шаблон:Mvar is its own inverse:

<math>{\operatorname{id}_X}^{-1} = \operatorname{id}_X.</math>

More generally, a function Шаблон:Math is equal to its own inverse, if and only if the composition Шаблон:Math is equal to Шаблон:Math. Such a function is called an involution.

Graph of the inverse

Файл:Inverse Function Graph.png

The graphs of Шаблон:Math and Шаблон:Math. The dotted line is Шаблон:Math.

If Шаблон:Mvar is invertible, then the graph of the function

is the same as the graph of the equation

This is identical to the equation Шаблон:Math that defines the graph of Шаблон:Mvar, except that the roles of Шаблон:Mvar and Шаблон:Mvar have been reversed. Thus the graph of Шаблон:Math can be obtained from the graph of Шаблон:Mvar by switching the positions of the Шаблон:Mvar and Шаблон:Mvar axes. This is equivalent to reflecting the graph across the line Шаблон:Math.^[16]^[1]

Inverses and derivatives

The inverse function theorem states that a continuous function Шаблон:Mvar 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

is invertible, since the derivative Шаблон:Math is always positive.

If the function Шаблон:Mvar is differentiable on an interval Шаблон:Mvar and Шаблон:Math for each Шаблон:Math, then the inverse Шаблон:Math is differentiable on Шаблон:Math.^[17] If Шаблон:Math, the derivative of the inverse is given by the inverse function theorem,

<math>\left(f^{-1}\right)^\prime (y) = \frac{1}{f'\left(x \right)}. </math>

Using Leibniz's notation the formula above can be written as

This result follows from the chain rule (see the article on inverse functions and differentiation).

The inverse function theorem can be generalized to functions of several variables. Specifically, a differentiable multivariable function Шаблон:Math is invertible in a neighborhood of a point Шаблон:Mvar as long as the Jacobian matrix of Шаблон:Mvar at Шаблон:Mvar is invertible. In this case, the Jacobian of Шаблон:Math at Шаблон:Math is the matrix inverse of the Jacobian of Шаблон:Mvar at Шаблон:Mvar.

Real-world examples

Let Шаблон:Mvar be the function that converts a temperature in degrees Celsius to a temperature in degrees Fahrenheit, <math display="block"> F = f(C) = \tfrac95 C + 32 ;</math> then its inverse function converts degrees Fahrenheit to degrees Celsius, <math display="block"> C = f^{-1}(F) = \tfrac59 (F - 32) ,</math>^[18] since <math display="block">

\begin{align} f^{-1} (f(C)) = {} & f^{-1}\left( \tfrac95 C + 32 \right) = \tfrac59 \left( (\tfrac95 C + 32 ) - 32 \right) = C, \\ & \text{for every value of } C, \text{ and } \\[6pt] f\left(f^{-1}(F)\right) = {} & f\left(\tfrac59 (F - 32)\right) = \tfrac95 \left(\tfrac59 (F - 32)\right) + 32 = F, \\ & \text{for every value of } F. \end{align} </math>

Suppose Шаблон:Mvar 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, <math display="block">\begin{align}

f(\text{Allan})&=2005 , \quad & f(\text{Brad})&=2007 , \quad & f(\text{Cary})&=2001 \\
f^{-1}(2005)&=\text{Allan} , \quad & f^{-1}(2007)&=\text{Brad} , \quad & f^{-1}(2001)&=\text{Cary}

\end{align} </math>

Let Шаблон:Mvar be the function that leads to an Шаблон:Mvar percentage rise of some quantity, and Шаблон:Mvar be the function producing an Шаблон:Mvar percentage fall. Applied to $100 with Шаблон:Mvar = 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 Шаблон:Math. In many cases we need to find the concentration of acid from a pH measurement. The inverse function Шаблон:Math is used.

Generalizations

Partial inverses

Файл:Inverse square graph.svg

The square root of Шаблон:Mvar is a partial inverse to Шаблон:Math.

Even if a function Шаблон:Mvar is not one-to-one, it may be possible to define a partial inverse of Шаблон:Mvar by restricting the domain. For example, the function

is not one-to-one, since Шаблон:Math. However, the function becomes one-to-one if we restrict to the domain Шаблон:Math, in which case

(If we instead restrict to the domain Шаблон:Math, then the inverse is the negative of the square root of Шаблон:Mvar.) Alternatively, there is no need to restrict the domain if we are content with the inverse being a multivalued function:

Файл:Inversa d'una cúbica gràfica.png

The inverse of this cubic function has three branches.

Sometimes, this multivalued inverse is called the full inverse of Шаблон:Mvar, and the portions (such as Шаблон:Sqrt and −Шаблон:Sqrt) 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 Шаблон:Mvar is called the principal value of Шаблон:Math.

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 adjacent picture).

Файл:Gràfica del arcsinus.png

The arcsine is a partial inverse of the sine function.

These considerations are particularly important for defining the inverses of trigonometric functions. For example, the sine function is not one-to-one, since

for every real Шаблон:Mvar (and more generally Шаблон:Math for every integer Шаблон:Mvar). However, the sine is one-to-one on the interval Шаблон:Closed-closed, and the corresponding partial inverse is called the arcsine. This is considered the principal branch of the inverse sine, so the principal value of the inverse sine is always between −Шаблон:Sfrac and Шаблон:Sfrac. The following table describes the principal branch of each inverse trigonometric function:^[19]

function	Range of usual principal value
arcsin	Шаблон:Math
arccos	Шаблон:Math
arctan	Шаблон:Math
arccot	Шаблон:Math
arcsec	Шаблон:Math
arccsc	Шаблон:Math

Left and right inverses

Function composition on the left and on the right need not coincide. In general, the conditions

"There exists Шаблон:Mvar such that Шаблон:Math" and
"There exists Шаблон:Mvar such that Шаблон:Math"

imply different properties of Шаблон:Mvar. For example, let Шаблон:Math denote the squaring map, such that Шаблон:Math for all Шаблон:Mvar in Шаблон:Math, and let Шаблон:Math denote the square root map, such that Шаблон:Math Шаблон:Radic for all Шаблон:Math. Then Шаблон:Math for all Шаблон:Mvar in Шаблон:Closed-open; that is, Шаблон:Mvar is a right inverse to Шаблон:Mvar. However, Шаблон:Mvar is not a left inverse to Шаблон:Mvar, since, e.g., Шаблон:Math.

Left inverses

If Шаблон:Math, a left inverse for Шаблон:Mvar (or retraction of Шаблон:Mvar ) is a function Шаблон:Math such that composing Шаблон:Mvar with Шаблон:Mvar from the left gives the identity function^[20] <math display="block">g \circ f = \operatorname{id}_X\text{.}</math> That is, the function Шаблон:Mvar satisfies the rule

If Шаблон:Math, then Шаблон:Math.

The function Шаблон:Mvar must equal the inverse of Шаблон:Mvar on the image of Шаблон:Mvar, but may take any values for elements of Шаблон:Mvar not in the image.

A function Шаблон:Mvar with nonempty domain is injective if and only if it has a left inverse.^[21] An elementary proof runs as follows:

If Шаблон:Mvar is the left inverse of Шаблон:Mvar, and Шаблон:Math, then Шаблон:Math.
If nonempty Шаблон:Math is injective, construct a left inverse Шаблон:Math as follows: for all Шаблон:Math, if Шаблон:Mvar is in the image of Шаблон:Mvar, then there exists Шаблон:Math such that Шаблон:Math. Let Шаблон:Math; this definition is unique because Шаблон:Mvar is injective. Otherwise, let Шаблон:Math be an arbitrary element of Шаблон:Mvar.
For all Шаблон:Math, Шаблон:Math is in the image of Шаблон:Mvar. By construction, Шаблон:Math, the condition for a left inverse.

In classical mathematics, every injective function Шаблон:Mvar with a nonempty domain necessarily has a left inverse; however, this may fail in constructive mathematics. For instance, a left inverse of the inclusion Шаблон:Math of the two-element set in the reals violates indecomposability by giving a retraction of the real line to the set Шаблон:Math.^[22]

Right inverses

Файл:Right inverse with surjective function.svg

Example of right inverse with non-injective, surjective function

A right inverse for Шаблон:Mvar (or section of Шаблон:Mvar ) is a function Шаблон:Math such that

<math>f \circ h = \operatorname{id}_Y . </math>

That is, the function Шаблон:Mvar satisfies the rule

If <math>\displaystyle h(y) = x</math>, then <math>\displaystyle f(x) = y .</math>

Thus, Шаблон:Math may be any of the elements of Шаблон:Mvar that map to Шаблон:Mvar under Шаблон:Mvar.

A function Шаблон:Mvar has a right inverse if and only if it is surjective (though constructing such an inverse in general requires the axiom of choice).

If Шаблон:Mvar is the right inverse of Шаблон:Mvar, then Шаблон:Mvar is surjective. For all <math>y \in Y</math>, there is <math>x = h(y)</math> such that <math>f(x) = f(h(y)) = y</math>.

If Шаблон:Mvar is surjective, Шаблон:Mvar has a right inverse Шаблон:Mvar, which can be constructed as follows: for all <math>y \in Y</math>, there is at least one <math>x \in X</math> such that <math>f(x) = y</math> (because Шаблон:Mvar is surjective), so we choose one to be the value of Шаблон:Math.Шаблон:Citation needed

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 <math>g</math> is a left inverse and <math>h</math> a right inverse of <math>f</math>, for all <math>y \in Y</math>, <math>g(y) = g(f(h(y)) = h(y)</math>.

A function has a two-sided inverse if and only if it is bijective.

A bijective function Шаблон:Mvar is injective, so it has a left inverse (if Шаблон:Mvar is the empty function, <math>f \colon \varnothing \to \varnothing</math> is its own left inverse). Шаблон:Mvar is surjective, so it has a right inverse. By the above, the left and right inverse are the same.

If Шаблон:Mvar has a two-sided inverse Шаблон:Mvar, then Шаблон:Mvar is a left inverse and right inverse of Шаблон:Mvar, so Шаблон:Mvar is injective and surjective.

Preimages

If Шаблон:Math is any function (not necessarily invertible), the preimage (or inverse image) of an element Шаблон:Math is defined to be the set of all elements of Шаблон:Mvar that map to Шаблон:Mvar:

<math>f^{-1}(\{y\}) = \left\{ x\in X : f(x) = y \right\} . </math>

The preimage of Шаблон:Mvar can be thought of as the image of Шаблон:Mvar under the (multivalued) full inverse of the function Шаблон:Mvar.

Similarly, if Шаблон:Mvar is any subset of Шаблон:Mvar, the preimage of Шаблон:Mvar, denoted <math>f^{-1}(S) </math>, is the set of all elements of Шаблон:Mvar that map to Шаблон:Mvar:

<math>f^{-1}(S) = \left\{ x\in X : f(x) \in S \right\} . </math>

For example, take the function Шаблон:Math. This function is not invertible as it is not bijective, but preimages may be defined for subsets of the codomain, e.g.

<math>f^{-1}(\left\{1,4,9,16\right\}) = \left\{-4,-3,-2,-1,1,2,3,4\right\}</math>.

The preimage of a single element Шаблон:Math – a singleton set Шаблон:Math – is sometimes called the fiber of Шаблон:Mvar. When Шаблон:Mvar is the set of real numbers, it is common to refer to Шаблон:Math as a level set.

Notes

Шаблон:Reflist

References

Шаблон:Reflist

Bibliography

External links

Шаблон:Sister project links

Шаблон:Springer

Шаблон:Authority control

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[Peirce_1852-4] Ошибка цитирования Неверный тег <ref>; для сносок Peirce_1852 не указан текст

[Peano_1903-5] Ошибка цитирования Неверный тег <ref>; для сносок Peano_1903 не указан текст

[Cajori_1929-6] 6,0 ^6,1 ^6,2 ^6,3 Ошибка цитирования Неверный тег <ref>; для сносок Cajori_1929 не указан текст

[8] Helmut Sieber und Leopold Huber: Mathematische Begriffe und Formeln für Sekundarstufe I und II der Gymnasien. Ernst Klett Verlag.

[9] Шаблон:Harvnb

[Korn_2000-10] 9,0 ^9,1 Ошибка цитирования Неверный тег <ref>; для сносок Korn_2000 не указан текст

[Atlas_2009-11] 10,0 ^10,1 ^10,2 ^10,3 ^10,4 Ошибка цитирования Неверный тег <ref>; для сносок Atlas_2009 не указан текст

[Hall_1909-12] Ошибка цитирования Неверный тег <ref>; для сносок Hall_1909 не указан текст

[13] Шаблон:Harvnb

[Wolf72-14] 13,0 ^13,1 Шаблон:Harvnb

[15] Шаблон:Harvnb

[16] Шаблон:Harvnb

[17] Шаблон:Harvnb

[18] Шаблон:Harvnb

[:1-19] Шаблон:Cite web

[20] Шаблон:Harvnb

[21] Шаблон:Cite book

[22] Шаблон:Cite book

[23] Шаблон:Cite journal

[1]

[2]

[3]

[4]

[5]

[6]

[nb 1]

[7]

[8]

[9]

[10]

[11]

[12]

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