Inverse function

From CryptoDox, The Online Encyclopedia on Cryptography and Information Security

In mathematics, if ƒ is a function from A to B then an inverse function for ƒ, denoted by ƒ⁻¹, is a function in the opposite direction, from B to A, with the property that a round trip (a composition) returns each element to itself. 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:

Failed to parse (unknown error\tfrac): f(C) = \tfrac95 C + 32 ; \,\!

then its inverse function converts degrees Fahrenheit to degrees Celsius:

Failed to parse (unknown error\tfrac): 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,

Failed to parse (unknown error\begin): \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 1 Definition and notation 1.1 Inverses and codomains 1.2 Inverses and composition 1.3 Note on notation 2 Properties 2.1 Symmetry 2.2 Inverse of a composition 2.3 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

Definition and notation

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:

Failed to parse (unknown error\text): \text{If }y = f(x)\text{, then }f^{-1}(y) = x\text{.}\,\!

For this rule to define a function, each element y ∈ Y must correspond to exactly one element x ∈ X. A function ƒ with this property is called one-to-one or an injection. This inverse, if it exists, is unique.

Inverses and codomains

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. 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 Template:Mbox 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 Template:Mbox corresponds to exactly one element Template:Mbox.

Inverses and composition

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

Failed to parse (unknown error\begin): \begin{align} \text{1. }&f^{-1}\left( \, f(x) \, \right) = x\text{, for every }x \in X\text{, and} \\ \text{2. }&f\left( \, f^{-1}(y) \, \right) = y\text{, for every }y \in Y\text{.} \end{align}

These two statements are equivalent to the definition of the inverse. Using the composition of functions we can rewrite these statements as follows:

Failed to parse (unknown error\begin): \begin{align} \text{1. }&f^{-1} \circ f = \mathrm{id}_X\text{, and} \\ \text{2. }&f \circ f^{-1} = \mathrm{id}_Y\text{,} \end{align}

where id_X and id_Y are the identity functions on the sets X and Y. In category theory, these statements are used as the definition of an inverse morphism.

If we think of composition as a kind of multiplication of functions, these identities say 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 functions.

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 Template:Mbox, then Template:Mbox, or x + 2.

In calculus, ƒ⁽ⁿ⁾, with parentheses, denotes the nth derivative of a function ƒ.

In trigonometry, for historical reasons, sin²(x) usually does mean the square of sin(x). For instance, the expressions

Failed to parse (unknown error\text): \sin^2 x \quad \text{and}\quad (\sin x)^2

represent the same thing, the first being a convenient abbreviation for the second. However, the expressions

Failed to parse (unknown error\text): \sin^{-1} x \quad \text{and}\quad (\sin x)^{-1}

are different. The first denotes the inverse to the sine function (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:

The function Template:Mbox is the multiplicative inverse to the sine, and is called the cosecant. It is usually denoted csc x:

Properties

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. This can be 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 Template:Mbox, and let Template:Mbox. Then the composition Template:Mbox 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:

Failed to parse (unknown error\tfrac): (f \circ g)^{-1}(y) = \tfrac13(y - 5)

This is the composition Template:Mbox.

Self-inverses

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

More generally, a function Template:Mbox is equal to its own inverse if and only if the composition Template:Mbox is equal to id_x. 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. 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.e. as long as the graph of the function passes the horizontal line test.

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 Template:Mbox for x. For example, if ƒ is the function

then we must solve the equation Template:Mbox for x:

Failed to parse (unknown error\begin): \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

Failed to parse (unknown error\dfrac): 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

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 Template:Mbox 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 Template:Mbox 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 Template:Mbox.

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

is invertible, since the derivative Template:Mbox is always positive.

If the function ƒ is differentiable, then the inverse ƒ^–1 will be differentiable as long as Template:Mbox. The derivative of the inverse is given by the inverse function theorem:

If we set Template:Mbox, then the formula above can be written

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 function Template:Mbox 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.

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

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

(If we instead restrict to the domain Template:Mbox, 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 principle branch, and its value at y is called the principle 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, the sine function is not one-to-one, since

for every real x (and more generally Template:Mbox for every integer n). However, the sine is one-to-one on the interval Template:Mbox, and the corresponding partial inverse is called the arcsine. This is considered the principle branch of the inverse sine, so the principle value of the inverse sine is always between –^π⁄₂ and ^π⁄₂. The following table describes the principle 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 Template:Mbox such that

That is, the function g satisfies the rule

Failed to parse (unknown error\text): \text{If }f(x) = y\text{, 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 Template:Mbox such that

That is, the function h satisfies the rule

Failed to parse (unknown error\text): \text{If }h(y) = x\text{, 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.

Preimages

If ƒ: X → Y is any function (not necessarily invertible), the preimage (or inverse image) of an element Template:Mbox 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.

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 Template:Mbox 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.

See also

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

References

• Template:Citationio:Simetra elemento