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. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function In Mathematics, a composite function represents the application of one function to the results of another 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 degrees Fahrenheit:

then its inverse function converts degrees Fahrenheit to degrees Celsius:

Or, suppose ƒ assigns each child in a family of three the year of its birth. The Celsius Temperature scale was previously known as the centigrade scale. Fahrenheit is a temperature scale named after Daniel Gabriel Fahrenheit (1686–1736 a German Physicist who proposed it in 1724 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,

Contents 1 Definitions 1.1 Inverses in higher mathematics 1.2 Inverses and composition 1.3 Note on notation 2 Properties 2.1 Uniqueness 2.2 Symmetry 2.3 Inverse of a composition 2.4 Self-inverses 3 Inverses in calculus 3.1 Formula for the inverse 3.2 Graph of the inverse 3.3 Inverses and derivatives 4 Generalizations 4.1 Partial inverses 4.2 Left and right inverses 4.3 Preimages 5 See also 6 References

Definitions

Let ƒ be a function whose domain is the set X, and whose range is the set Y. In Mathematics, the domain of a given function is the set of " Input " values for which the function is defined In Mathematics, the range of a function is the set of all "output" values produced by that function Then the inverse of ƒ is the function ƒ^–1 with domain Y and range X, defined by the following rule:

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.

For instance, if ƒ(x) = y = x², 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. In Algebra, the square of a number is that number multiplied by itself 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 Mathematics, a square root of a number x is a number r such that r 2 = x, or in words a number r whose In Mathematics, a square root of a number x is a number r such that r 2 = x, or in words a number r whose 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. Calculus ( Latin, calculus, a small stone used for counting is a branch of Mathematics that includes the study of limits, Derivatives In higher mathematics, the notation

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. In Mathematics, the codomain, or target, of a function f: X → Y is the set The codomain contains the range of ƒ as a subset, and is considered part of the definition of ƒ.

When using codomains, the inverse of a function 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. In Mathematics, a function f is said to be surjective or onto, if its values span its whole Codomain; that is for every In Mathematics, a function f is said to be surjective or onto, if its values span its whole Codomain; that is for every 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 y ∈ Y corresponds to exactly one element x ∈ X. In Mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property

Inverses and composition

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

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:

where id_X is the identity function on the set X. In Mathematics, a composite function represents the application of one function to the results of another In Mathematics, an identity function, also called identity map or identity transformation, is a function that always returns the same value that In category theory, this statement is used as the definition of an inverse morphism. In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets In Mathematics, a morphism is an abstraction derived from structure-preserving mappings between two Mathematical structures The study of morphisms and

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. In Mathematics, a multiplicative inverse for a number x, denoted by 1&frasl x or x &minus1 is a number which 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. In Mathematics, the hyperbolic functions are analogs of the ordinary trigonometric, or 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. The dynamical system concept is a mathematical Formalization for any fixed "rule" which describes the Time dependence of a point's position In Mathematics, iterated functions are the objects of deep study in Computer science, Fractals and Dynamical systems An iterated function is 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 calculus, ƒ⁽ⁿ⁾, with parentheses, denotes the nth derivative of a function ƒ. In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change For instance:

In trigonometry, for historical reasons, sin²(x) usually does mean the square of sin(x):

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

It denotes the inverse function for sin(x) (actually a partial inverse; see below). Circle-trig6svg|300px|thumb|right|All of the Trigonometric functions of an angle θ can be constructed geometrically in terms of a unit circle centered at O. To avoid confusion, an inverse trigonometric function is often indicated by the prefix "arc". For instance the inverse sine is typically called the arcsine:

The function (sin x)^–1 is the multiplicative inverse to the sine, and is called the cosecant. It is usually denoted 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:

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:

Inverse of a composition

The inverse of a composition of functions is given by the formula

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:

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

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:

More generally, a function ƒ: X → X is equal to its own inverse if and only if the composition ƒ o ƒ is equal to id_x. In Mathematics, an identity function, also called identity map or identity transformation, is a function that always returns the same value that Such a function is called an involution.

Inverses in calculus

Single-variable calculus is primarily concerned with functions that map real numbers to real numbers. Calculus ( Latin, calculus, a small stone used for counting is a branch of Mathematics that includes the study of limits, Derivatives In Mathematics, the real numbers may be described informally in several different ways Such functions are often defined through formulas, such as:

A function ƒ from the real numbers to the real numbers possesses an inverse as long as it is one-to-one, i. In Mathematics and in the Sciences a formula (plural formulae, formulæ or formulas) is a concise way of expressing information 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	y – a
a – x	a – y
mx	y / m	m ≠ 0
1 / x	1 / y	x, y ≠ 0
x2		x, y ≥ 0 only, in general
x3		no restriction on x and y
xp	y1/p (i. e. )	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)

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

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

Thus the inverse function ƒ^–1 is given by the formula

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

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

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

is the same as the graph of the equation

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 Mathematics, 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 strictly increasing or decreasing (with no local maxima or minima). In Mathematics, a continuous function is a function for which intuitively small changes in the input result in small changes in the output In Mathematics, maxima and minima, known collectively as extrema, are the largest value (maximum or smallest value (minimum that For example, the function

is invertible, since the derivative ƒ′(x) = 3x² + 1 is always positive. In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change

If the function ƒ is differentiable, then the inverse ƒ^–1 will be differentiable as long as ƒ′(x) ≠ 0. In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change The derivative of the inverse is given by the inverse function theorem:

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

This result follows from the chain rule (see the article on inverse functions and differentiation). In Mathematics, the inverse function theorem gives sufficient conditions for a Vector-valued function to be Invertible on an Open region containing In Calculus, the chain rule is a Formula for the Derivative of the composite of two functions. In Mathematics, the inverse of a function y = f(x is a function that in some fashion "undoes" the effect of f (see Inverse

The inverse function theorem can be generalized to functions of several variables. Specifically, a differentiable function ƒ: Rⁿ → Rⁿ is invertible in a neighborhood of a point p as long as the Jacobian matrix of ƒ at p is invertible. In Vector calculus, the Jacobian is shorthand for either the Jacobian matrix or its Determinant, the Jacobian determinant. In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally In Linear algebra, an n -by- n (square matrix A is called invertible or non-singular if there exists an n -by- In this case, the Jacobian of ƒ^–1 at ƒ(p) is the matrix inverse of the Jacobian of ƒ at p. In Linear algebra, an n -by- n (square matrix A is called invertible or non-singular if there exists an n -by-

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. The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function For example, the function

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

(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:

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. In Mathematics, maxima and minima, known collectively as extrema, are the largest value (maximum or smallest value (minimum that 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). This article discusses cubic equations in one variable For a discussion of cubic equations in two variables see Elliptic curve.

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 x (and more generally sin(x + 2πn) = sin(x) for every integer n). The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French However, the sine is one-to-one on the interval [–^π⁄₂, ^π⁄₂], 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 –^π⁄₂ and ^π⁄₂. The following table describes the principal branch of each inverse trigonometric function:

function	Range of usual principal value
sin–1	–π⁄2 ≤ sin–1(x) ≤ π⁄2
cos–1	0 ≤ cos–1(x) ≤π
tan–1	–π⁄2 < tan–1(x) < π⁄2
cot–1	0 < cot–1(x) < π
sec–1	0 < sec–1(x) < π
csc–1	−π⁄2 ≤ csc–1(x) < π⁄2

Left and right inverses

If ƒ: X → Y, a left inverse for ƒ (or retraction of ƒ) is a function g: Y → X such that

That is, the function g satisfies the rule

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. In considering complex Multiple-valued functions in Complex analysis, the principal values of a function are the values along one chosen branch of that In Category theory, a branch of Mathematics, a section is a right inverse of a morphism 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: Y → X such that

That is, the function h satisfies the rule

Thus, h(y) may be any of the elements of x that map to y under ƒ. In Category theory, a branch of Mathematics, a section is a right inverse of a morphism 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: X → Y is any function (not necessarily invertible), the preimage (or inverse image) of an element y ∈ Y is the set of all elements of X that map to y:

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 a preimage under a given function is the set of all possible function outputs when taking each element of the preimage

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

The preimage of a single element y ∈ Y is sometimes called the fiber of y. In Mathematics, the fiber of a point y under a function f    X  →  Y is the inverse When Y is the set of real numbers, it is common to refer to ƒ^–1(y) as a level set. In Mathematics, a level set of a real -valued function f of n variables is a set of the form { ( x 1

See also

• Inverse trigonometric function
• Logarithm
• Inverse function theorem
• Inverse functions and differentiation
• Inverse relation
• Inverse element

References

• Smith, William K. In Mathematics, the logarithm of a number to a given base is the power or Exponent to which the base must be raised in order to produce In Mathematics, the inverse function theorem gives sufficient conditions for a Vector-valued function to be Invertible on an Open region containing In Mathematics, the inverse of a function y = f(x is a function that in some fashion "undoes" the effect of f (see Inverse In Mathematics, the inverse relation of a Binary relation is the relation taken 'backwards' as in changing the relation 'child of' to 'parent of' In Mathematics, the idea of inverse element generalises the concepts of negation, in relation to Addition, and reciprocal, in relation to Inverse Functions, MacMillan, 1966.
• Stewart, James (2002), Calculus (5th ed. ), Brooks Cole, ISBN 978-0534393397 

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