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Encyclopedia > Inverse function
A function ƒ and its inverse ƒ–1. Because ƒ maps a to 3, the inverse ƒ–1 maps 3 back to a.
A function ƒ and its inverse ƒ–1. Because ƒ maps a to 3, the inverse ƒ–1 maps 3 back to a.

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. Not every function has an inverse; those that do are called invertible. For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... This article is about functions in mathematics. ... In mathematics, a composite function, formed by the composition of one function on another, represents the application of the former to the result of the application of the latter to the argument of the composite. ...


For example, let ƒ be the function that converts a temperature in degrees Celsius to a temperature in degrees Fahrenheit: For other uses, see Celsius (disambiguation). ... For other uses, see Fahrenheit (disambiguation). ...

 f(C) = tfrac95 C + 32 ; ,!

then its inverse function converts degrees Fahrenheit to degrees Celsius:

 f^{-1}(F) = tfrac59 (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,

begin{align} f(text{Alan})&=2005 , quad & f(text{Bree})&=2007 , quad & f(text{Cary})&=2001  f^{-1}(2001)&=text{Cary} , quad & f^{-1}(2005)&=text{Alan} , quad & f^{-1}(2007)&=text{Bree} end{align}

Contents

Definitions

If ƒ maps X to Y, then ƒ–1 maps Y back to X.
If ƒ maps X to Y, then ƒ–1 maps Y back to X.

Let ƒ be a function whose domain is the set X, and whose range is the set Y. Then the inverse of ƒ is the function ƒ–1 with domain Y and range X, defined by the following rule: In mathematics, the domain of a function is the set of all input values to the function. ... In mathematics, a set can be thought of as any collection of distinct objects considered as a whole. ... In mathematics, the range of a function is the set of all output values produced by that function. ...

text{If }f(x) = ytext{, then }f^{-1}(y) = xtext{.},!

Thus, an inverse function uniquely identifies the input x of another function based only on its output y, for all y ∈ Y. Not all functions have an inverse. For this rule to be appliable, each element y ∈ Y must correspond to exactly one element x ∈ X. A function ƒ with this property is called one-to-one, or information-preserving, or an injection. One-to-one redirects here. ...


For instance, if ƒ(x) = y = x2, each element in Y would correspond to two different elements in Xx), 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 the principal square root function is the inverse of x2 because the first is not single-valued, and the second returns -x when x is negative. In algebra, the square of a number is that number multiplied by itself. ... In mathematics, a square root (√) of a number x is a number r such that , or in words, a number r whose square (the result of multiplying the number by itself) is x. ... A single-valued function is an emphatic term for a mathematical function in the usual sense. ...


Inverses in higher mathematics

The definition given above is commonly adopted in calculus. In higher mathematics, the notation For other uses, see Calculus (disambiguation). ...

fcolon X to 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 the codomain. The codomain contains the range of ƒ as a subset, and is considered part of the definition of ƒ. A codomain in mathematics is the set of output values associated with (or mapped to) the domain of inputs in a function. ... Superset redirects here. ...


When using codomains, the inverse of a function ƒ: XY 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 a bijection, and has the property that every element yY corresponds to exactly one element xX. A surjective function. ... A surjective function. ... ↔ ⇔ ≡ logical symbols representing iff. ... A bijective function. ...


Inverses and composition

If ƒ is an invertible function with domain X and range Y, then

 f^{-1}left( , f(x) , right) = xtext{, for every }x in Xtext{.}

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: In mathematics, a composite function, formed by the composition of one function on another, represents the application of the former to the result of the application of the latter to the argument of the composite. ...

 f^{-1} circ f = mathrm{id}_Xtext{,}

where idX is the identity function on the set X. In category theory, this statement is used as the definition of an inverse morphism. An identity function f is a function which doesnt have any effect: it always returns the same value that was used as its argument. ... In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them. ... In mathematics, a morphism is an abstraction of a structure-preserving process between two mathematical structures. ...


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. The reciprocal function: y = 1/x. ...


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. In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ... A ray through the origin intercepts the hyperbola in the point , where is the area between the ray, its mirror image with respect to the -axis, and the hyperbola (see animated version with comparison with the trigonometric (circular) 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 an exponent. A similar notation is used in dynamical systems for iterated functions. For example, ƒ2 denotes two iterations of the function ƒ; if ƒ(x) = x + 1, then ƒ2(x) = (x + 1) + 1, or x + 2. In symbols: In mathematics, exponentiation is a process generalized from repeated multiplication, in much the same way that multiplication is a process generalized from repeated addition. ... The Lorenz attractor is an example of a non-linear dynamical system. ... In mathematics, iterated functions are the objects of study in fractals and dynamical systems. ...

f^2(x) = f(f(x)) = (f circ f)(x).

In calculus, ƒ(n), with parentheses, denotes the nth derivative of a function ƒ. For instance: This article is about derivatives and differentiation in mathematical calculus. ...

f^{(2)}(x) = frac{d^{2}}{dx^{2}}f(x).

In trigonometry, for historical reasons, sin2(x) usually does mean the square of sin(x): Wikibooks has a book on the topic of Trigonometry The Canadarm2 robotic manipulator on the International Space Station is operated by controlling the angles of its joints. ...

 sin^2 x = (sin x)^2. ,!

However, the expression sin-1(x) does not represent the multiplicative inverse to sin(x):

sin^{-1} x neq (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 the arcsine: In mathematics, the inverse trigonometric functions are the inverse functions of the trigonometric functions. ... In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ...

sin^{-1} x = arcsin x = mathrm{asin}, x. ,!

The function (sin x)–1 is the multiplicative inverse to the sine, and is called the cosecant. It is usually denoted csc x: In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ...

(sin x)^{-1} = frac{1}{sin x} = csc x . ,!

Properties

Uniqueness

If an inverse function exists for a given function ƒ, it is unique.


Symmetry

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:

begin{align} &text{If } &f^{-1} circ f = mathrm{id}_Xtext{,}  &text{then } &f circ f^{-1} = mathrm{id}_Ytext{.} end{align}

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}right)^{-1} = f . ,!

Inverse of a composition

The inverse of g o ƒ is ƒ–1 o g–1.
The inverse of g o ƒ is ƒ–1 o g–1.

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 ƒ(x) = x + 5, and let g(x) = 3x. Then the composition 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) = tfrac13(y - 5)

This is the composition (g–1 o ƒ–1) (y).


Self-inverses

If X is a set, then the identity function on X is its own inverse: An identity function f is a function which doesnt have any effect: it always returns the same value that was used as its argument. ...

mathrm{id}_X^{-1} = mathrm{id}_X

More generally, a function ƒ: XX is equal to its own inverse if and only if the composition ƒ o ƒ is equal to idx. Such a function is called an involution. In mathematics, an involution is a function that is its own inverse, so that f(f(x)) = x for all x in the domain of f. ...


Inverses in calculus

Single-variable calculus is primarily concerned with functions that map real numbers to real numbers. Such functions are often defined through formulas, such as: For other uses, see Calculus (disambiguation). ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics and in the sciences, a formula (plural: formulae, formulæ or formulas) is a concise way of expressing information symbolically (as in a mathematical or chemical formula), or a general relationship between quantities. ...

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 the horizontal line test. In mathematics, the horizontal line test is a test used to determine if a function is injective, surjective or bijective. ...


The following table shows several standard functions and their inverses:

Function ƒ(x) Inverse ƒ–1(y) Notes
x + a ya
ax ay
mx y / m m ≠ 0
1 / x 1 / y x, y ≠ 0
x2 sqrt{y} x, y ≥ 0 only, pmsqrt{y} in general
x3 sqrt[3]{y} no restriction on x and y
xp y1/p (i.e. sqrt[p]{y}) x, y ≥ 0 in general, p ≠ 0
ex ln y y > 0
ax loga y y > 0 and a > 0
trigonometric functions inverse trigonometric functions various restrictions (see table below)

Sine redirects here. ... In mathematics, the inverse trigonometric functions are the inverse functions of the trigonometric functions. ...

Formula for the inverse

One approach to finding a formula for ƒ–1, if it exists, is to solve the equation y = ƒ(x) for x. For example, if ƒ is the function

f(x) = (2x + 8)^3 ,!

then we must solve the equation y = (2x + 8)3 for x:

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}

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 y = x + sin x cannot be solved algebraically for x.


Graph of the inverse

The graphs of y = ƒ(x) and y = ƒ–1(x). The dotted line is y = x.
The graphs of y = ƒ(x) and y = ƒ–1(x). The dotted line is y = x.

If ƒ and ƒ–1 are inverses, then the graph of the function

is the same as the graph of the equation

x = f(y) . ,!

This is identical to the equation 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 line y = x. In erik, a reflection (also spelled reflexion) is a map that transforms an object into its mirror image. ...


Inverses and derivatives

A continuous function ƒ is one-to-one (and hence invertible) if and only if it is either increasing or decreasing (with no local maxima or minima). For example, the function In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ... A monotonically increasing function (it is strictly increasing on the left and just non-decreasing on the right). ... Local and global maxima and minima for cos(3Ï€x)/x, 0. ...

is invertible, since the derivative ƒ′(x) = 3x2 + 1 is always positive. This article is about derivatives and differentiation in mathematical calculus. ...


If the function ƒ is differentiable, then the inverse ƒ–1 will be differentiable as long as ƒ′(x) ≠ 0. The derivative of the inverse is given by the inverse function theorem: In mathematics, the derivative of a function is one of the two central concepts of calculus. ... In mathematics, the inverse function theorem gives sufficient conditions for a vector-valued function to be invertible on an open region containing a point in its domain. ...

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

If we set x = ƒ–1(y), then the formula above can be written

frac{dx}{dy} = frac{1}{dy / dx} .

This result follows from the chain rule (see the article on inverse functions and differentiation). In calculus, the chain rule is a formula for the derivative of the composite of two functions. ... In mathematics, the inverse of a function is a function that, in some fashion, undoes the effect of (see inverse function for a formal and detailed definition). ...


The inverse function theorem can be generalized to functions of several variables. Specifically, a differentiable function ƒ: RnRn is invertible in a neighborhood of a point p as long as the Jacobian matrix of ƒ at p is invertible. In this case, the Jacobian of ƒ–1 at ƒ(p) is the matrix inverse of the Jacobian of ƒ at p. For the French Revolution faction, see Jacobin. ... In mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries), which may be numbers or, more generally, any abstract quantities that can be added and multiplied. ... In linear algebra, an n-by-n (square) matrix A is called invertible or non-singular if there exists an n-by-n matrix B such that where In denotes the n-by-n identity matrix and the multiplication used is ordinary matrix multiplication. ... In mathematics and especially linear algebra, an n-by-n matrix A is called invertible, non-singular or regular if there exists another n-by-n matrix B such that AB = BA = In, where In denotes the n-by-n identity matrix and the multiplication used is ordinary matrix multiplication. ...


Generalizations

Partial inverses

The square root of x is a partial inverse to ƒ(x) = x2.
The square root of x is a partial inverse to ƒ(x) = x2.

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 This article is about functions in mathematics. ...

f(x) = x^2,!

is not one-to-one, since x2 = (–x)2. However, the function becomes one-to-one if we restrict to the domain x ≥ 0, in which case

f^{-1}(y) = sqrt{y} .

(If we instead restrict to the domain 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: This diagram does not represent a true function, because the element 3 in X is associated with two elements, b and c, in Y. In mathematics, a multivalued function is a total relation; i. ...

f^{-1}(y) = pmsqrt{y} .
The inverse of this cubic function has three branches.
The inverse of this cubic function has three branches.

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). Polynomial of degree 3 In mathematics, a cubic function is a function of the form where b is nonzero; or in other words, a polynomial of degree three. ...


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). Polynomial of degree 3 In mathematics, a cubic function is a function of the form where b is nonzero; or in other words, a polynomial of degree three. ...

The arcsine is a partial inverse of the sine function.
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 In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ... In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ... In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ... In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ...

sin(x + 2pi) = sin(x),!

for every real x (and more generally sin(x + 2πn) = sin(x) for every integer n). However, the sine is one-to-one on the interval [–π2, π2], 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 –π2 and π2. The following table describes the principal branch of each inverse trigonometric function: The integers are commonly denoted by the above symbol. ... In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ...

function Range of usual principal value
sin–1 π2 ≤ sin–1(x) ≤ π2
cos–1 0 ≤ cos–1(x) ≤π
tan–1 π2 < tan–1(x) < π2
cot–1 0 < cot–1(x) < π
sec–1 0 < sec–1(x) < π
csc–1 π2 ≤ csc–1(x) < π2

See also Cauchy principal value for its use in describing improper integrals In considering complex multiple-valued functions in complex analysis, the principal values of a function are the values along one chosen branch of that function, so it is single-valued. ...

Left and right inverses

If ƒ: XY, a left inverse for ƒ (or retraction of ƒ) is a function g: YX such that

g circ f = mathrm{id}_X . ,!

That is, the function g satisfies the rule

text{If }f(x) = ytext{, 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 h: YX such that In the mathematical field of category theory, a section is a morphism which has a left inverse, i. ...

f circ h = mathrm{id}_Y . ,!

That is, the function h satisfies the rule

text{If }h(y) = xtext{, 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). In mathematics, the axiom of choice, or AC, is an axiom of set theory. ...


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.


Preimages

If ƒ: XY is any function (not necessarily invertible), the preimage (or inverse image) of an element yY is the set of all elements of X that map to y:

f^{-1}(y) = left{ xin X : f(x) = y right} . ,!

The preimage of y can be thought of as the image of y under the (multivalued) full inverse of the function f. In mathematics, the image of an element x in a set X under the function f : X → Y, denoted by f(x), is the unique y in Y that is associated with x. ...


Similarly, if S is any subset of Y, the preimage of S is the set of all elements of X that map to S: Superset redirects here. ...

f^{-1}(S) = left{ xin X : f(x) in S right} . ,!

The preimage of a single element yY 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. In mathematics, the fiber of a point y under a function f : X â†’ Y is the inverse image of y under f, that is, f–1(y). ... In mathematics, a level set of a real-valued function f of n variables is a set of the form { (x1,...,xn) | f(x1,...,xn) = c } where c is a constant. ...


See also

In mathematics, the inverse trigonometric functions are the inverse functions of the trigonometric functions. ... Look up logarithm in Wiktionary, the free dictionary. ... In mathematics, the inverse function theorem gives sufficient conditions for a vector-valued function to be invertible on an open region containing a point in its domain. ... In mathematics, the inverse of a function is a function that, in some fashion, undoes the effect of (see inverse function for a formal and detailed definition). ... In logic and mathematics, the inverse relation of a binary relation is the binary relation defined by . ... In mathematics, the inverse of an element x, with respect to an operation *, is an element x such that their compose gives a neutral element. ...

References

  • Smith, William K. Inverse Functions, MacMillan, 1966.
  • Stewart, James (2002), Calculus (5th ed.), Brooks Cole, ISBN 978-0534393397

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