Inverse function

File:Inverse Function.png

A function Template:Mvar and its inverse f⁻¹. Because Template:Mvar maps Template:Mvar to 3, the inverse f⁻¹ maps 3 back to Template:Mvar.

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

For a function $f:X\to Y$, its inverse $f^{-1}:Y\to X$ admits an explicit description: it sends each element $y\in Y$ to the unique element $x\in X$ such that f(x) = y.

As an example, consider the real-valued function of a real variable given by f(x) = 5x − 7. One can think of Template: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 Template:Mvar is the function $f^{-1}:\mathbb{ℝ}\to \mathbb{ℝ}$ defined by $f^{-1}(y)=\frac{y+7}{5}.$

Definitions

File:Inverse Functions Domain and Range.png

If Template:Mvar maps Template:Mvar to Template:Mvar, then f⁻¹ maps Template:Mvar back to Template:Mvar.

Let Template:Mvar be a function whose domain is the set Template:Mvar, and whose codomain is the set Template:Mvar. Then Template:Mvar is invertible if there exists a function Template:Mvar from Template:Mvar to Template:Mvar such that $g(f(x))=x$ for all $x\in X$ and $f(g(y))=y$ for all $y\in Y$.

If Template:Mvar is invertible, then there is exactly one function Template:Mvar satisfying this property. The function Template:Mvar is called the inverse of Template:Mvar, and is usually denoted as f⁻¹, a notation introduced by John Frederick William Herschel in 1813.

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

The inverse function f⁻¹ to Template:Mvar can be explicitly described as the function

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

Inverses and composition

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

$f^{-1}(f(x))=x$, for every $x\in X$ and $f(f^{-1}(y))=y$ for every $y\in Y$.

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

$f^{-1}\circ f={\mathrm{id}}_X$ and $f\circ f^{-1}={\mathrm{id}}_Y,$

where id_X is the identity function on the set Template: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 f⁻¹. Repeatedly composing a function f: X→X with itself is called iteration. If Template:Mvar is applied Template:Mvar times, starting with the value Template:Mvar, then this is written as fⁿ(x); so f²(x) = f (f (x)), etc. Since f⁻¹(f (x)) = x, composing f⁻¹ and fⁿ yields fⁿ⁻¹, "undoing" the effect of one application of Template:Mvar.

Notation

While the notation f⁻¹(x) might be misunderstood, (f(x))⁻¹ certainly denotes the multiplicative inverse of f(x) and has nothing to do with the inverse function of Template:Mvar. The notation $f^{⟨-1⟩}$ might be used for the inverse function to avoid ambiguity with the multiplicative inverse.

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

Examples

Squaring and square root functions

The function f: R → [0,∞) given by f(x) = x² is not injective because $(-x{)}^2=x^2$ for all $x\in \mathbb{ℝ}$. Therefore, Template:Mvar is not invertible.

If the domain of the function is restricted to the nonnegative reals, that is, we take the function $f:[0,\mathrm{\infty })\to [0,\mathrm{\infty });x\mapsto x^2$ with the same rule as before, then the function is bijective and so, invertible. The inverse function here is called the (positive) square root function and is denoted by $x\mapsto \sqrt{x}$.

Standard inverse functions

The following table shows several standard functions and their inverses:

Inverse arithmetic functions

Function f(x)	Inverse f −1(y)	Notes
x + a	y − a
a − x	a − y
mx	Template:Sfrac	m ≠ 0
Template:Sfrac (i.e. x−1)	Template:Sfrac (i.e. y−1)	x, y ≠ 0
xp	$\sqrt[p]{y}$ (i.e. y1/p)	integer p > 0; x, y ≥ 0 if p is even
ax	loga y	y > 0 and a > 0 and a ≠ 1
xex	W (y)	x ≥ −1 and y ≥ −1/e
trigonometric functions	inverse trigonometric functions	various restrictions (see table below)
hyperbolic functions	inverse hyperbolic functions	various restrictions
logistic function	logit

Formula for the inverse

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

$f^{-1}(y)=(\text{the unique element }x\in \mathbb{ℝ}\text{ such that }f(x)=y)$.

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

$f(x)=(2x+8{)}^3$

then to determine $f^{-1}(y)$ for a real number Template:Mvar, one must find the unique real number Template:Mvar such that (2x + 8)³ = y. This equation can be solved:

$\begin{array}{rl}y& =(2x+8{)}^3\\ \sqrt[3]{y}& =2x+8\\ \sqrt[3]{y}-8& =2x\\ \frac{\sqrt[3]{y}-8}{2}& =x.\end{array}$

Thus the inverse function f⁻¹ is given by the formula

$f^{-1}(y)=\frac{\sqrt[3]{y}-8}{2}.$

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

$f(x)=x-\sin x,$

then Template:Mvar is a bijection, and therefore possesses an inverse function f⁻¹. The formula for this inverse has an expression as an infinite sum:

$f^{-1}(y)=\sum _{n=1}^{\mathrm{\infty }}\frac{y^{n/3}}{n!}\underset{\theta \to 0}{\lim }\left(\frac{{\mathrm{d}}^{n-1}}{\mathrm{d}{\theta }^{n-1}}{\left(\frac{\theta }{\sqrt[3]{\theta -\sin (\theta )}}\right)}^n\right).$

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 Template:Mvar, then it is unique. This follows since the inverse function must be the converse relation, which is completely determined by Template:Mvar.

Symmetry

There is a symmetry between a function and its inverse. Specifically, if Template:Mvar is an invertible function with domain Template:Mvar and codomain Template:Mvar, then its inverse f⁻¹ has domain Template:Mvar and image Template:Mvar, and the inverse of f⁻¹ is the original function Template:Mvar. In symbols, for functions f:X → Y and f⁻¹:Y → X,

$f^{-1}\circ f={\mathrm{id}}_X$ and $f\circ f^{-1}={\mathrm{id}}_Y.$

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

${\left(f^{-1}\right)}^{-1}=f.$

File:Composition of Inverses.png

The inverse of g ∘ f is f⁻¹ ∘ g⁻¹.

The inverse of a composition of functions is given by

$(g\circ f{)}^{-1}=f^{-1}\circ g^{-1}.$

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

For example, let f(x) = 3x and let g(x) = x + 5. Then the composition g ∘ f is the function that first multiplies by three and then adds five,

$(g\circ f)(x)=3x+5.$

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

$(g\circ f{)}^{-1}(x)=\frac{1}{3}(x-5).$

This is the composition (f⁻¹ ∘ g⁻¹)(x).

Self-inverses

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

${{\mathrm{id}}_X}^{-1}={\mathrm{id}}_X.$

More generally, a function f : X → X is equal to its own inverse, if and only if the composition f ∘ f is equal to id_X. Such a function is called an involution.

Graph of the inverse

File:Inverse Function Graph.png

The graphs of y = f(x) and y = f⁻¹(x). The dotted line is y = x.

If Template:Mvar is invertible, 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 y = f(x) that defines the graph of Template:Mvar, except that the roles of Template:Mvar and Template:Mvar have been reversed. Thus the graph of f⁻¹ can be obtained from the graph of Template:Mvar by switching the positions of the Template:Mvar and Template:Mvar axes. This is equivalent to reflecting the graph across the line y = x.

Inverses and derivatives

By the inverse function theorem, a continuous function of a single variable $f:A\to \mathbb{ℝ}$ (where $A\subseteq \mathbb{ℝ}$) 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

$f(x)=x^3+x$

is invertible, since the derivative f′(x) = 3x² + 1 is always positive.

If the function Template:Mvar is differentiable on an interval Template:Mvar and f′(x) ≠ 0 for each x ∈ I, then the inverse f⁻¹ is differentiable on f(I). If y = f(x), the derivative of the inverse is given by the inverse function theorem,

${\left(f^{-1}\right)}^{\prime }(y)=\frac{1}{f^{\prime }\left(x\right)}.$

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

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

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 continuously differentiable multivariable function f : Rⁿ → Rⁿ is invertible in a neighborhood of a point Template:Mvar as long as the Jacobian matrix of Template:Mvar at Template:Mvar is invertible. In this case, the Jacobian of f⁻¹ at f(p) is the matrix inverse of the Jacobian of Template:Mvar at Template:Mvar.

Real-world examples

• Let Template:Mvar be the function that converts a temperature in degrees Celsius to a temperature in degrees Fahrenheit, $$F=f(C)=\frac{9}{5}C+32;$$ then its inverse function converts degrees Fahrenheit to degrees Celsius, $$C=f^{-1}(F)=\frac{5}{9}(F-32),$$ since $$\begin{array}{rl}{f}^{-1}(f(C))=& f^{-1}(\frac{9}{5}C+32)=\frac{5}{9}((\frac{9}{5}C+32)-32)=C,\\ & \text{for every value of }C,\text{ and }\\ f(f^{-1}(F))=& f(\frac{5}{9}(F-32))=\frac{9}{5}(\frac{5}{9}(F-32))+32=F,\\ & \text{for every value of }F.\end{array}$$
• Suppose Template: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, $$\begin{array}{rlrlrl}f(\text{Allan})& =2005,& f(\text{Brad})& =2007,& f(\text{Cary})& =2001\\ f^{-1}(2005)& =\text{Allan},& f^{-1}(2007)& =\text{Brad},& f^{-1}(2001)& =\text{Cary}\end{array}$$
• Let Template:Mvar be the function that leads to an Template:Mvar percentage rise of some quantity, and Template:Mvar be the function producing an Template:Mvar percentage fall. Applied to $100 with Template: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 pH = −log₁₀[H⁺]. In many cases we need to find the concentration of acid from a pH measurement. The inverse function [H⁺] = 10^−pH is used.

Generalizations

Partial inverses

File:Inverse square graph.svg

The square root of Template:Mvar is a partial inverse to f(x) = x².

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

$f(x)=x^2$

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 Template:Mvar.)

Full inverses

File:Inversa d'una cúbica gràfica.png

The inverse of this cubic function has three branches.

Alternatively, there is no need to restrict the domain if we are content with the inverse being a multivalued function:

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

Sometimes, this multivalued inverse is called the full inverse of Template:Mvar, and the portions (such as

1. REDIRECT Template:Radic

Template:Rcat shell and −

1. REDIRECT Template:Radic

Template:Rcat shell) 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 Template:Mvar is called the principal value of f⁻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 adjacent picture).

Trigonometric inverses

File:Gràfica del arcsinus.png

The arcsine is a partial inverse of the sine function.

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

$\sin (x+2\pi )=\sin (x)$

for every real Template:Mvar (and more generally sin(x + 2Template:Pin) = sin(x) for every integer Template:Mvar). However, the sine is one-to-one on the interval Template: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 −Template:Sfrac and Template:Sfrac. The following table describes the principal branch of each inverse trigonometric function:

function	Range of usual principal value
arcsin	−Template:Sfrac ≤ sin−1(x) ≤ Template:Sfrac
arccos	0 ≤ cos−1(x) ≤ Template:Pi
arctan	−Template:Sfrac < tan−1(x) < Template:Sfrac
arccot	0 < cot−1(x) < Template:Pi
arcsec	0 ≤ sec−1(x) ≤ Template:Pi
arccsc	−Template:Sfrac ≤ csc−1(x) ≤ Template:Sfrac

Left and right inverses

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

1. "There exists Template:Mvar such that g(f(x))=x" and
2. "There exists Template:Mvar such that f(g(x))=x"

imply different properties of Template:Mvar. For example, let f: R → Template:Closed-open denote the squaring map, such that f(x) = x² for all Template:Mvar in R, and let Template:Mvar: Template:Closed-open → R denote the square root map, such that g(x) = Template:Radic for all x ≥ 0. Then f(g(x)) = x for all Template:Mvar in Template:Closed-open; that is, Template:Mvar is a right inverse to Template:Mvar. However, Template:Mvar is not a left inverse to Template:Mvar, since, e.g., g(f(−1)) = 1 ≠ −1.

Left inverses

If f: X → Y, a left inverse for Template:Mvar (or retraction of Template:Mvar ) is a function g: Y → X such that composing Template:Mvar with Template:Mvar from the left gives the identity function $$g\circ f={\mathrm{id}}_X\text{.}$$ That is, the function Template:Mvar satisfies the rule

If f(x)=y, then g(y)=x.

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

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

• If Template:Mvar is the left inverse of Template:Mvar, and f(x) = f(y), then g(f(x)) = g(f(y)) = x = y.

• If nonempty f: X → Y is injective, construct a left inverse g: Y → X as follows: for all y ∈ Y, if Template:Mvar is in the image of Template:Mvar, then there exists x ∈ X such that f(x) = y. Let g(y) = x; this definition is unique because Template:Mvar is injective. Otherwise, let g(y) be an arbitrary element of Template:Mvar.

  For all x ∈ X, f(x) is in the image of Template:Mvar. By construction, g(f(x)) = x, the condition for a left inverse.

In classical mathematics, every injective function Template: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 {0,1} → R of the two-element set in the reals violates indecomposability by giving a retraction of the real line to the set {0,1}.

Right inverses

File:Right inverse with surjective function.svg

Example of right inverse with non-injective, surjective function

A right inverse for Template:Mvar (or section of Template:Mvar ) is a function h: Y → X such that

$f\circ h={\mathrm{id}}_Y.$

That is, the function Template:Mvar satisfies the rule

If $h(y)=x$, then $f(x)=y.$

Thus, h(y) may be any of the elements of Template:Mvar that map to Template:Mvar under Template:Mvar.

A function Template:Mvar has a right inverse if and only if it is surjective (this equivalence holds if, and only if, the axiom of choice holds).

If Template:Mvar is the right inverse of Template:Mvar, then Template:Mvar is surjective. For all $y\in Y$, there is $x=h(y)$ such that $f(x)=f(h(y))=y$.

If Template:Mvar is surjective, Template:Mvar has a right inverse Template:Mvar, which can be constructed as follows: for all $y\in Y$, there is at least one $x\in X$ such that $f(x)=y$ (because Template:Mvar is surjective), so we choose one to be the value of h(y).

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

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

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

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

Preimages

If f: X → Y is any function (not necessarily invertible), the preimage (or inverse image) of an element y ∈ Y is defined to be the set of all elements of Template:Mvar that map to Template:Mvar:

$f^{-1}(y)=\{x\in X:f(x)=y\}.$

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

The notion can be generalized to subsets of the range. Specifically, if Template:Mvar is any subset of Template:Mvar, the preimage of Template:Mvar, denoted by $f^{-1}(S)$, is the set of all elements of Template:Mvar that map to Template:Mvar:

$f^{-1}(S)=\{x\in X:f(x)\in S\}.$

For example, take the function f: R → R; x ↦ x². This function is not invertible as it is not bijective, but preimages may be defined for subsets of the codomain, e.g.

$f^{-1}(\{1,4,9,16\})=\{-4,-3,-2,-1,1,2,3,4\}$.

The original notion and its generalization are related by the identity $f^{-1}(y)=f^{-1}(\{y\}),$ The preimage of a single element y ∈ Y – a singleton set {y}  – is sometimes called the fiber of Template:Mvar. When Template:Mvar is the set of real numbers, it is common to refer to f⁻¹({y}) as a level set.

See also

• Lagrange inversion theorem, gives the Taylor series expansion of the inverse function of an analytic function
• Integral of inverse functions
• Inverse Fourier transform
• Reversible computing

External links

• Template:Springer