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:

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. 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,

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:

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 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 corresponds to exactly one element .

Inverses and composition

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

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

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 , then , 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

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

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 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 , and let . Then the composition 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 .

Self-inverses

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

More generally, a function is equal to its own inverse if and only if the composition 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 for x. For example, if ƒ is the function

then we must solve the equation 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 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 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 .

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 is always positive.

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

If we set , 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 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 . However, the function becomes one-to-one if we restrict to the domain , in which case

(If we instead restrict to the domain , 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. 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 for every integer n). However, the sine is one-to-one on the interval , 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 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. 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 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. 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 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 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

• id="CITEREFStewart">Stewart, James (2002), Calculus (5th ed.), Brooks Cole, ISBN 978-0534393397
  Mathematics (colloquially, maths or math) is the body of knowledge centered on such concepts as quantity, structure, space, and change, and also the academic discipline that studies them. Benjamin Peirce called it "the science that draws necessary conclusions".
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  function expresses dependence between two quantities, one of which is given (the independent variable, argument of the function, or its "input") and the other produced (the dependent variable, value of the function, or "output").
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  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.
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  Celsius is, or relates to, the Celsius temperature scale (previously known as the centigrade scale). The degree Celsius (symbol: °C) can refer to a specific temperature on the Celsius scale
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  Fahrenheit is a temperature scale named after the German-Dutch physicist Daniel Gabriel Fahrenheit (1686–1736), who proposed it in 1724.
  
  In this scale, the melting point of water is 32 degrees Fahrenheit (written “32 °F”), and the boiling point is
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  domain is most often defined as the set of values, D for which a function is defined.^[1] A function that has a domain N is said to be a function over N, where N is an arbitrary set.
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  SET may stand for:

  • Sanlih Entertainment Television, a television channel in Taiwan
  • Secure electronic transaction, a protocol used for credit card processing,

  
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  In mathematics, the range of a function is the set of all "output" values produced by that function. Sometimes it is called the image, or more precisely, the image of the domain of the function.
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  Injection may refer to:

  • Injection (medicine), a method of putting liquid into the body with a syringe and a hollow needle that punctures the skin.
  • Injective function in mathematics, a function which associates distinct arguments to distinct values

  
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  In mathematics, the codomain of a function : → is the set .
  
  The domain of is the set .
  
  The range of is the set defined as .
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  subset of a set B if A is "contained" inside B. Notice that A and B may coincide. The relationship of one set being a subset of another is called inclusion or containment.
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  non-surjective function.]] In mathematics, a function f is said to be surjective if its values span its whole codomain; that is, for every y in the codomain, there is at least one x in the domain such that f(x) = y .
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  non-surjective function.]] In mathematics, a function f is said to be surjective if its values span its whole codomain; that is, for every y in the codomain, there is at least one x in the domain such that f(x) = y .
  ..... Click the link for more information.
  In mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property that, for every y in Y, there is exactly one x in X such that
  f(x) = y.
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  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.
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  In mathematics, an identity function, also called identity map or identity transformation, is a function that always returns the same value that was used as its argument.
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  In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them. Categories now appear in most branches of mathematics and in some areas of theoretical computer science and mathematical physics, and have been a unifying notion.
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  In mathematics, a morphism is an abstraction of a structure-preserving mapping between two mathematical structures.
  
  The most common example occurs when the process is a function or map which preserves the structure in some sense.
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  multiplicative inverse for a number x, denoted by ¹⁄_x or x ⁻¹, is a number which when multiplied by x yields the multiplicative identity, 1.
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  trigonometric functions (also called circular functions) are functions of an angle. They are important in the study of triangles and modeling periodic phenomena, among many other applications.
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  dynamical system concept is a mathematical formalization for any fixed "rule" which describes the time dependence of a point's position in its ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and
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  In mathematics, iterated functions are the objects of deep study in fractals and dynamical systems. An iterated function is a function which is composed with itself, repeatedly, a process called iteration.
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  derivative is a measurement of how a function changes when the values of its inputs change. Loosely speaking, a derivative can be thought of as how much a quantity is changing at some given point.
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  Trigonometry (from Greek trigōnon "triangle" + metron "measure"^[1]), informally called trig, is a branch of mathematics that deals with triangles, particularly triangles in a plane where one angle of the triangle is 90 degrees (right angled
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  In mathematics, the inverse trigonometric functions are the inverse functions of the trigonometric functions. The principal inverses are listed in the following table.
  
  Name Usual notation Definition Domain of x for real result Range of usual principal value
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  In mathematics, the inverse trigonometric functions are the inverse functions of the trigonometric functions. The principal inverses are listed in the following table.
  
  Name Usual notation Definition Domain of x for real result Range of usual principal value
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  trigonometric functions (also called circular functions) are functions of an angle. They are important in the study of triangles and modeling periodic phenomena, among many other applications.
  ..... Click the link for more information.
  In mathematics, an identity function, also called identity map or identity transformation, is a function that always returns the same value that was used as its argument.
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  involution, or an involutary function, is a function that is its own inverse, so that
  
  

  f(f(x)) = x for all x in the domain of f.

  General properties

  Any involution is a bijection.
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  Calculus (Latin, calculus, a small stone used for counting) is a branch of mathematics that includes the study of limits, derivatives, integrals, and infinite series, and constitutes a major part of modern university education.
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