Inverse function

In mathematics, the inverse function of a bijective function is the function that assigns to each element of the target set its uniquely determined original element.

function $f\colon A\to B$ assigns to each $a\in A$ uniquely determined element $b \in B$ , f(a) denoted by If the relation holds for $a\in A,b\in B$ $b=f(a)$ then it is also said that is a primal element of under general, an element of may have no, one or more primal elements under If each element of has exactly one primal element under (this is called the primal element), called invertible. In this case, one can define a function $f^{-1}\colon B \to A$ that assigns to each element of its uniquely defined primal element under This function is then called the inverse function of

It is easy to prove that a function is invertible if it is bijective (i.e. simultaneously injective and surjective). In fact, injectivity says nothing else than that every element of has at most one primal element under Surjectivity just says that every element of has at least one primal element under }.

The term inverse function formally belongs to the mathematical subfield of set theory, but is used in many subfields of mathematics.

The inverse function

Definition

Let and be non-empty sets. Besides the definition from the introduction, there are other ways to formally introduce the notions of invertibility of a function $f\colon A\to B$ and the inverse function of an invertible function:

One searches for a function $g\colon B \to A$ such that $g(f(a))=a$ for all $a\in A$ and $f(g(b))=b$ for all $b \in B$ . It turns out that there can be at most one such If this exists, then called invertible and the uniquely determined called the inverse function of .
Using the composition of functions, the previous condition can also be formulated more elegantly by $g\colon B \to A$ requiring for : $g \circ f = \operatorname{id}_A$ and $f\circ g=\operatorname {id} _{B}$ . Where $\operatorname {id} _{A}$ the identical mapping on the set .
One first introduces the concepts of left inverses and right inverses explained below. Then a function is called invertible if it has both a left inverse and a right inverse. It can be seen that in this case left inverses and right inverses must coincide (with which it also follows that in this case there are not several of them). This uniquely determined left and right inverse is then the inverse function.
The definition refers to the fact that a function from to is always also a relation from to Therefore, it has an inverse relation in any case. One calls invertible if this inverse relation is a function from to In this case, the inverse relation is also called an inverse function.

It turns out that all the invertibility notions presented are equivalent to the notion of bijectivity. Also, all definitions of the inverse function lead to the same result.

Simple examples

Let be $A:=\{a,b,c,\dotsc ,y,z\}$ the set of 26 letters of the Latin alphabet and let $B:=\{1,2,3,\dotsc ,25,26\}$ . The function $f \colon A \rightarrow B$ which assigns to each letter the corresponding number in the alphabet is bijective, and $f^{-1}\colon B\rightarrow A$ is given by $f^{-1}(n) =$ "the nth letter in the alphabet".

Let $f \colon \mathbb{R} \rightarrow \mathbb{R}$ the real function with . This is bijective and the inverse function is given by

$f^{-1}\colon \mathbb {R} \to \mathbb {R} ,\quad f^{-1}(x)=(x-2)/3$ .

More generally, if α $\alpha ,\beta \in \mathbb {R}$ and the function $f \colon \mathbb{R} \rightarrow \mathbb{R}$ given by $f(x)=\alpha x+\beta$ . Then is bijective exactly α $\alpha \neq 0$ . In this case $f^{-1}(x)={\tfrac {x-\beta }{\alpha }}$ .

Properties

The inverse function is itself bijective. Its inverse function is the original function, i.e.

$( f^{-1} )^{-1} = f$ .

If $f \colon A \rightarrow B$ a bijective function, then the inverse function holds:

$f(f^{-1}(b))=b$ for all $b \in B$ ,

$f^{-1}(f(a))=a$ for all $a\in A$ .

Or a little more elegantly:

$f \circ f^{-1} = \operatorname{id}_B$

$f^{-1} \circ f = \operatorname{id}_A$ .

If $f \colon A \rightarrow B$ and are $g \colon B \rightarrow A$ two functions with the property

$f(g(b))=b$ for all $b \in B$

then it can already be concluded from each of the three following properties that both functions are bijective and their mutual inverse functions:

$g(f(a))=a$ for all $a\in A$ ,

is injective

is surjective

If the functions $f\colon A\rightarrow B$ and are $g\colon B \rightarrow C$ bijective, then this is also true for the composition $g \circ f : A \rightarrow C$ . The inverse function of $g\circ f$ is then $f^{-1} \circ g^{-1}$ .

A function $f\colon A \rightarrow A$ can be its own inverse function. This is true exactly when $f \circ f = \operatorname{id}_A$ . In this case, called an involution. The simplest involutory mappings are the identical mappings.

If $f\colon A\rightarrow B$ a bijective function, where and subsets of $\mathbb {R}$ , then the graph of the inverse function is obtained by mirroring the graph of on the straight line

If $f \colon \mathbb{R} \rightarrow \mathbb{R}$ differentiable, $f'(x) \neq 0$ and then the following inverse rule applies:

$(f^{-1})'(y)=\frac{1}{f'(f^{-1}(y))}$ .

This statement is generalised in multidimensional analysis to the theorem of inverse mapping.

Inverse function for non-bijective functions

In many cases there is a desire for an inverse function for a non-bijective function. The following tools can be used for this purpose:

If the function is not surjective, one can reduce the target set by choosing just the image of the function for this purpose. The function obtained in this way is surjective and agrees in its course with the original function. This approach is always possible. However, it may be difficult to determine exactly the image of the function under consideration. In addition, an important property of the originally considered target set can be lost in the transition to this subset (in calculus, for example, completeness).

In some cases it also proves fruitful to achieve the desired surjectivity by extending the definition range of the function under consideration. This is often accompanied by an expansion of the target set. Whether this path is feasible and meaningful must be decided individually in each case.

If the function is not injective, one can define a suitable equivalence relation on its domain of definition so that one can transfer the function to the set of corresponding equivalence classes. This function is then automatically injective. However, this approach is demanding and leads to an often undesirable change in the nature of the arguments of the function under consideration.

In practice, one can often also achieve injectivity of the function by restricting to a suitable subset of the function's domain of definition that contains only a single primal element for each element of the image. However, this restriction may be arbitrary. One must therefore ensure that this restriction is made consistently in the same way at all points.

Examples

Consider the successor function $n\mapsto n+1$ on the set $\mathbb {N}$ of the natural numbers without the zero. This function is injective. However, it is not surjective because the number 1 does not occur as a function value. One can now remove the number 1 from the target set. Then the function becomes surjective and the predecessor function $n\mapsto n-1$ is its inverse function. However, it is unattractive that the function's definition range and target set no longer correspond.

The alternative idea of extending the definition range by the missing original element for the 1, namely the 0, has the same disadvantage at first glance. If, in order to remedy this, 0 is also added to the target set, this again has no original element. However, one can continue this process mentally infinitely often and thereby arrive at the set $\mathbb {Z}$ of the integers. On this set, the successor function is bijective and its inverse function is the predecessor function.

The exponential function considered as a function from $\mathbb {R}$ to $\mathbb {R}$ is injective but not surjective. Its image is just the set of positive real numbers. If one restricts the target set to this, one obtains a bijective function whose inverse function is the logarithm function. A natural extension of the number range, as discussed in the previous example, does not lend itself here. Therefore, one must accept that for the functions under consideration, the definition range and the target set no longer coincide.

The square function $x\mapsto x^{2}$ is considered as a function from $\mathbb {R}$ to $\mathbb {R}$ neither injective nor surjective. Surjectivity is achieved by taking as the target set the image set $\mathbb{R}_0^+ = [ 0, \infty )$ of the non-negative real numbers. To achieve injectivity, one can restrict the domain of definition. The most obvious one here is also to take $\mathbb {R} _{0}^{+}$ . The restricted square function obtained in this way is bijective. Its inverse function is the square root function.

The trigonometric functions sine (sin), cosine (cos) and tangent (tan) are not bijective. One restricts in each case to suitable subsets of the domain of definition and the target set and obtains bijective functions whose inverse functions are the arcsine functions: Arc sine (arcsin), arc cosine (arccos) and arc tangent (arctan).

A corresponding procedure for the hyperbolic functions sine hyperbolicus (sinh), cosine hyperbolicus (cosh) and tangent hyperbolicus (tanh) leads to the area functions: Areasinus hyperbolicus (arsinh), Areakosinus hyperbolicus (arcosh) and Areatangens hyperbolicus (artanh).

Calculation

The effective determination of the inverse function is often difficult. Asymmetric encryption methods are based on the fact that determining the inverse function of an encryption function is effectively only possible if a secret key is known. In this case, the calculation rule for the encryption function itself is publicly known.

Real functions are often defined by a calculation rule that can be described by an arithmentic term (with a variable ). In the search for the inverse function one now tries to convert the function equation $y=T(x)$ by equivalent transformation into the form $x=T'(y)$ (for a suitable term ), i.e. to resolve equivalently to . If this succeeds, the function defined by the calculation rule proved to be bijective and is a calculation rule for the inverse function. Note that in the steps of the equivalence transformation, the sets from which and to be chosen must be carefully considered. They then form the domain of definition and the target set of the function under consideration.

Examples:

Let $f\colon \mathbb {R} \to \mathbb {R}$ with . The following equations are equivalent:

${\begin{aligned}y&=2x-1\\2x&=y+1\\x&={\tfrac {y+1}{2}}\end{aligned}}$

The inverse function of is therefore $f^{-1}(y)=\tfrac{y+1}{2}$ . Since it is usual to denote the argument by one also writes: $f^{-1}(x)=\tfrac{x+1}{2}$ .

Let $f\colon (0,\infty )\to \mathbb {R}$ with $f(x) = \tfrac{x^2-1}{2x}$ . The following equations are equivalent (note that

${\begin{aligned}&y={\tfrac {x^{2}-1}{2x}}\\&2xy=x^{2}-1\\&x^{2}-2xy-1=0\\&x=y+{\sqrt {y^{2}+1}}\end{aligned}}$

(The second solution of the quadratic equation is omitted because assumed to be positive). The inverse function is therefore: $f^{-1}(y) = y + \sqrt{y^2+1}$

Note: The square root was used in this solution. The square root function is just defined as the inverse function of the simple square function $x\mapsto x^{2}$ . This is because this simple function cannot be 'inverted' using basic arithmetic.

This problem was solved by adding another member (namely the square root) to the stock of standard mathematical operations.

The achievement of the transformation carried out above is thus to have traced the calculation for the inverse function of the function back to the calculation of the inverse function of the square function.

The square root, as I said, cannot be calculated in an elementary way. In fact, it often has irrational values even for integer arguments. However, there are well-understood approximation methods for the square root.

Therefore, the above transformation is considered sufficient. In fact, a better result cannot be achieved either.

Note that the other inverse functions given above (logarithm, arcus and area functions) cannot be calculated with the help of the basic arithmetic operations (and the exponential function and the trigonometric functions) either. Therefore, just like the square root, they expand the set of standard mathematical operations (see also elementary function).

Inverse functions and morphisms

In higher mathematics, sets are often considered that are still provided with additional mathematical structure. A simple example of this is the set of natural numbers, on which there is, among other things, the order structure defined by the Kleiner relation.

If we now consider functions between two sets that carry the same type of structure (e.g. two ordered sets), we are particularly interested in functions between these sets that are 'compatible' with the corresponding structures. This compatibility must be defined separately. However, the definition is obvious in most cases.

Functions that fulfil this compatibility are also called morphisms. For ordered sets, the morphisms are approximately the monotonic functions.

If a morphism is bijective, the question arises whether the inverse function is also a morphism.

In many areas of mathematics, this is automatically the case. For example, the inverse functions of bijective homomorphisms are automatically also homomorphisms.

This is not the case in other subfields. In the case of ordered sets, for example, it depends on whether one restricts oneself to total orders (then inverse functions of monotone functions are monotone again) or whether one also allows half orders (then this is not always the case).

A bijective morphism whose inverse function is also a morphism is also called an isomorphism.

Inverse functions of linear mappings

A particularly important example of the concept of morphism is the concept of linear mapping (the vector space homomorphism). A bijective linear mapping is always an isomorphism. The question often arises as to how its inverse function can be effectively determined.

For such an isomorphism to exist at all, the two vector spaces involved must have the same dimension. If this is finite, then every linear mapping between the spaces can be represented by a square matrix (with the corresponding number of columns). The linear mapping is then bijective if this matrix has an inverse. This inverse then describes the inverse function.

In the mathematical subfield of functional analysis, one primarily considers infinite-dimensional vector spaces that carry an additional topological structure in addition to the vector space structure. Only linear mappings that are also compatible with the topological structures, i.e. continuous, are accepted as morphisms. In general, the inverse function of a bijective continuous linear mapping between two topological vector spaces is not necessarily continuous. However, if both spaces involved are Banach spaces, it follows from the open mapping theorem that this must be the case.

Generalisations

For more general applications, the concept of the inverse function as the inverse of a bijection introduced above is too narrow. Accordingly, generalisations exist for such circumstances, two of which are presented below.

Left inverse

For a function a function $f\colon A\rightarrow B$ is called $g\colon B\rightarrow A$ left inverse (or retraction) if

$g\circ f=\mathrm {id} _{A}.\,\!$

That is, the function satisfies

${\text{Wenn }}f(a)=b{\text{, dann }}g(b)=a.\,\!$

The behaviour of on the image of is therefore fixed. For elements from which are not the result of can take on any value. A function has left inverses exactly when it is injective (left unambiguous).

An injective function can have several left inverses. This is exactly the case if the function is not surjective and the domain of definition has more than one element.

Examples

Left inverses often occur as 'inverses' of embeddings.

For example, let be the set of clubs that have a team in the first men's Bundesliga in the 2018/19 season. Let the set of municipalities in Germany. The function assigns a club the municipality in which its stadium is located. Since no two Bundesliga teams come from the same city in the season under consideration, this function is injective. Since there are also municipalities without a Bundesliga stadium, it is not surjective. Thus, there are several left inverses to . A simple link inverse to form is the function that assigns the associated club to every municipality that has a Bundesliga stadium and FC Bayern München to all other municipalities. A more practical example would be the function that assigns to each municipality the Bundesliga club with the closest stadium. However, it would also be much more time-consuming to determine this function, especially since it would first have to be clarified which concept of distance the definition is based on (as the crow flies, shortest distance by car, ...).

As a numerical example, let the embedding of $\mathbb {Z}$ in $\mathbb {R}$ . Then any rounding function (to 0 decimal places), for example the Gaussian bracket, lends itself as a left inverse. But also the function on $\mathbb {R}$ which assigns itself to every integer and 0 to all other numbers, is also a left inverse.

Right inverse

A right inverse (coretraction) of $f\colon A\rightarrow B$ (or, for fibre bundles, an intersection of ) is a function $h\colon B\rightarrow A$ such that

$f\circ h=\mathrm {id} _{B}.\,\!$

That is, the function satisfies

${\text{Wenn }}h(b)=a{\text{, dann }}f(a)=b.\,\!$

$h(b)$ can therefore be any primal element of under

If a function has a right inverse, it must be surjective (right total).

Conversely, it seems obvious that from the surjectivity of follows the existence of a right inverse. For every $b \in B$ one can find one or even more primal elements under in However, if the function is 'highly non-injective', a decision must be made for an unmanageable number of elements of the target set as to which of the primal elements one really takes in each case. Such a simultaneous decision cannot always be made constructively. The axiom of choice (in a suitable formulation) states that a right inverse nevertheless exists for all surjective functions.

In many cases, however, the ambiguity can be resolved by a global definition. This is the case, for example, with the definition of the square root, where the ambiguity is always resolved in favour of the positive solution. In such cases, the axiom of choice is not needed.

The function is obviously a right inverse of if is the left inverse of follows directly from this that right inverses are always injective and left inverses always surjective.

A surjective function has multiple right inverses exactly when it is not injective.

Examples

Right inverses often occur as functions that determine representatives of a set.

For example, let $f\colon \text{Art} \rightarrow \text{Gattung}$ be a function that assigns each species its genus. As a right inverse then chooses a function that names a typical species for each genus. Political representation provides many examples. Here could be the nationality of a person, the head of state of a country.

The Hilbert curve maps the unit interval continuously (hence the name curve) to the unit square. In practical applications, however, the Hilbert index is often needed, namely a linearisation of two-dimensional data (an inversion of the Hilbert curve). To do this, one takes one of the right inverses of the Hilbert curve, of which there are several - because the Hilbert curve, as a continuous mapping between two spaces of different dimensions, cannot be bijective according to the invariance of dimension theorem.

Left and right inverses of morphisms

If the sets and carry an additional mathematical structure and if $f\colon A\rightarrow B$ an injective or surjective function, respectively, which is compatible with these structures, the question arises whether it is possible to choose the left or right inverse, respectively, in such a way that it is also compatible with the structures. For many structures studied in mathematics, this is not the case. However,is an injective or surjective linear mapping, the left or right inverse can also be chosen as a linear mapping.

Miscellaneous

Of particular interest are often functions for which the domain of definition and the target set coincide. For a set the set of functions of forms a monoid in itself with the composition as a link. The notions of invertibility and left and right inverse introduced here then coincide with the corresponding notions from algebra.

The term inverse function in this case is identical to the term inverseelement.

In the general context, the notion of invertibility of functions is often omitted because it coincides with the notion of bijectivity.

In the above considerations it was assumed that and are non-empty. If empty, then there is only a function from to if also empty.

This is then the empty function that is bijective and involutory.

If but not , then there is again exactly one function from to which is also empty. This function is injective but not surjective. It has neither left nor right inverses, since there are no functions from to all.

There are different approaches to introducing the concept of function in mathematics. The concept of surjectivity used in this article assumes that the target set is part of the identity of the function. If a different concept of function is used as a basis, some of the explanations must be adapted accordingly.

Most of the statements in this article also apply to functions between classes.

Inverse function

Definition

Simple examples

Properties

Inverse function for non-bijective functions

Examples

Calculation

Inverse functions and morphisms

Inverse functions of linear mappings

Generalisations

Left inverse

Right inverse

Left and right inverses of morphisms

Miscellaneous

See also

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