A function assigns each a clearly defined element , which is referred to with. When it comes to the relationship , one also says that a prototype element is from below . In general, an element can have zero, one or more archetype elements under . If each element has exactly one archetype element below (one then speaks of the archetype element), it is called invertible . In this case, a function can be defined that assigns each element a clearly defined archetype element . This function is then referred to as the inverse of .
One can easily prove that a function is invertible if and only if it is bijective (i.e. simultaneously injective and surjective ). In fact, injectivity means nothing other than that each element has at most one archetype element . Surjectivity says that every element has at least one archetype element below .
The concept of the inverse function belongs formally to the mathematical sub-area of set theory , but is used in many sub-areas of mathematics.
Be and non-empty sets . In addition to the definition from the introduction, there are other ways to formally introduce the terms of the invertibility of a function and the inverse function of an invertible function:
- One looks for a function so that for everyone and for everyone . It turns out that there can be at most one such . If this exists, it is called invertible and that uniquely determines the inverse function of .
- Using the composition of functions, the previous condition can also be formulated a little more elegantly by asking for : and . It is the identity function on the set .
- Initially passing the stated below terms of left inverse and right inverse one. Then a function is called invertible if it has both a left inverse and a right inverse. It turns out that in this case left inverse and right inverse must match (which also means that in this case there are not more than one). This clearly defined 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 definitely has an inverse relation . We call it invertible if this inverse relation is a function of to . In this case, the inverse relation is also called the inverse function.
It turns out that all the invertibility concepts presented are equivalent to the concept of bijectivity. All definitions of the inverse function also lead to the same result.
If is a bijective function, then denotes the inverse function. The superscript should not be confused with a negative power with regard to multiplication . Rather, it is the reverse of the composition of functions. The alternative spelling (f across) can easily be confused with the complex conjugation . It is therefore rarely used in the mathematical literature.
However, there is also an ambiguity in the notation . This notation is used for the archetype function that exists for every function (including non-bijective). The archetype function is a function of the power set into the power set . It is common in the notation of the archetype function to leave out the set brackets for single-element sets. So for is written simply instead of . If one now identifies the one-element set with the one contained element in this notational way, then the inverse function is a specialization of the archetype function, and frontal contradictions cannot arise. For for bijectives is the one and only element of the set of archetypes .
Because of the possibility of confusion mentioned, the notation for the inverse function (that is the -th iteration ) is occasionally found in the literature
so that (with raised pointed brackets),
and ( without superscript parentheses)
and for the derivation
and (with raised round bracket).
Then for example
- Let be the set of the 26 letters of the Latin alphabet and be . The function that assigns the corresponding number in the alphabet to each letter is bijective and is given by "the nth letter in the alphabet".
- Let be the real function with . This is bijective and the inverse function is given by
- More general: Are and given by the function . Then is bijective if and only if . In this case .
- The inverse function is itself bijective. Its inverse function is the original function; H.
- If it is a bijective function, then for the inverse function:
- for all ,
- for everyone .
- Or something more elegant:
- Are and two functions with the property
- for all
then from each of the three following properties it can already be concluded that both functions are bijective and their reciprocal inverse functions:
- for all ,
- is injective
- is surjective
- If the functions are and bijective, then this also applies to the composition . The inverse function of is then .
- A function can be its own inverse. This is true exactly when . In this case it is called an involution . The simplest involutor maps are identical maps .
- If a bijective function, where and are subsets of , then the graph of the inverse function is created by mirroring the graph of on the straight line .
- This statement is generalized 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, the target set can be reduced by choosing the function image for this purpose . The function obtained in this way is surjective and its course corresponds to the original function. This approach is always possible. However, it can be difficult to precisely determine the picture of the function under consideration. In addition, when switching to this subset, an important property of the originally considered target set can be lost (such as completeness in analysis ).
- In some cases it also proves to be fruitful to achieve the desired surjectivity by expanding the domain of definition for the function under consideration . This often goes hand in hand with an expansion of the target amount. Whether this path is feasible and sensible, however, has to be decided on an individual basis.
- If the function is not injective, a suitable equivalence relation can be defined on its domain , so that the function can be transferred to the set of the corresponding equivalence classes . This function is then automatically injective. However, this approach is demanding and often leads to an undesirable change in the nature of the arguments of the function under consideration.
- In practice, the injectivity of the function can often also be achieved by restricting oneself to a suitable subset of the definition range of the function that contains only a single original image element for each element of the image. However, this restriction may be arbitrary. You must therefore ensure that you apply this restriction consistently in the same way at all points.
- Consider the successor function on the set of natural numbers without the zero. This function is injective. But it is not surjective because the number 1 does not appear as a function value. You can now remove the number 1 from the target set. Then the function becomes surjective and the predecessor function is its inverse function. However, it is unpleasant that the definition range and target set no longer match in the function.
- The alternative idea of expanding the domain to include the missing archetype element for the 1, namely the 0, has the same disadvantage at first glance. If, in order to remedy this, the 0 is also added to the target set, then this in turn has no archetype element. But you can continue this process infinitely often and thereby arrive at the set of whole numbers . The successor function is bijective on this set, and its inverse function is the predecessor function.
- The exponential function viewed as a function from to is injective but not surjective. Your picture 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, is not an option here. Therefore, one has to accept that the domain of definition and target set no longer coincide with the functions under consideration.
- The square function is considered as a function from to neither injective nor surjective. Surjectivity is achieved by choosing the image set of non-negative real numbers as the target set . In order to achieve injectivity, one can restrict the definition range. The most obvious thing to do is to choose here as well . The restricted square function thus obtained is bijective. Its inverse function is the square root function .
- The trigonometric functions sine (sin), cosine (cos) and tangent (tan) are not bijective. It restricts respectively to appropriate subsets of the domain and the target quantity and receives bijective functions whose inverse functions of the Inverse Trigonometric Functions : arcsine (arcsin), arccosine (arccos) and arctangent (arctan) are.
- 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).
Determining the inverse function effectively is often difficult. Asymmetric encryption methods are based on the fact that the reverse function of an encryption function can only be effectively determined if one knows a secret key. 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 ). When searching for the inverse function one tries to bring the functional equation into the form (for a suitable term ) by equivalence transformation , i.e. to solve it equivalently . If this succeeds, the function defined by the calculation rule is proven to be bijective and is a calculation rule for the inverse function. It should be noted that in the steps of the equivalence transformation, the quantities from which and are to be selected must be carefully observed. You then form the domain and target set of the function under consideration.
- Be with . The following equations are equivalent:
- The inverse function of is therefore . Since it is common that argument refer to, it also writes .
- Be with . The following equations are equivalent (note that ):
- (The second solution to the quadratic equation is omitted, since it is assumed to be positive.) The inverse function is thus:
- Note: The square root was used in this solution. The square root function is just defined as the inverse of the simple square function . This simple function cannot be 'reversed' using the basic arithmetic operations.
- This problem was solved by adding another member (namely the square root) to the set of standard mathematical operations.
- So the achievement of the transformation performed above is to have reduced the calculation for the inverse function of the function to the calculation of the inverse function of the square function.
- As I said, the square root cannot be calculated in an elementary way. In fact, even for integer arguments, it often has irrational values. 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, arc and area functions) cannot be calculated using the basic arithmetic operations (and the exponential function and the trigonometric functions). They therefore, just like the square root, expand the set of standard mathematical operations (see also elementary function ).
Inverse functions and morphisms
In higher mathematics, sets are often considered that are 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 smaller relation .
If one looks now at functions between two sets that have the same type of structure (i.e. two ordered sets), one is particularly interested in functions between these sets that are 'compatible' with the corresponding structures. This compatibility must be defined separately. The definition is obvious in most cases.
If a morphism is bijective, the question arises whether the inverse function is also a morphism.
This is automatically the case in many areas of mathematics. For example, the inverse functions of bijective homomorphisms are automatically homomorphisms as well.
This is not the case in other sub-areas. In the case of ordered sets, for example, it depends on whether one restricts oneself to total orders (then inverse functions of monotonic functions are again monotonic) or whether one also allows partial 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 (vector space homomorphism). A bijective linear mapping is always an isomorphism. The question often arises how their inverse function can be effectively determined.
So that such an isomorphism can exist at all, the two vector spaces involved must have the same dimension . If this is finite, every linear mapping between the rooms can be represented by a square matrix (with the corresponding number of columns). The linear mapping is then bijective if and only if this matrix has an inverse . This inverse then describes the inverse function.
In the mathematical sub-area of functional analysis , one looks primarily at infinite-dimensional vector spaces, which in addition to the vector space structure also have an additional topological structure. Only linear mappings that are compatible with the topological structures, that is, continuous , are accepted as morphisms here . In general, the inverse function of a bijective continuous linear mapping between two topological vector spaces is not necessarily continuous. But if both spaces involved are Banach spaces , then it follows from the theorem about the open mapping that this must be the case.
The concept of the inverse function introduced above as the inverse of a bijection is too narrow for more general applications. Accordingly, generalizations exist for such situations, two of which are presented below.
That is, it fulfills the function
So the behavior of in the picture of is fixed. For elements from that are not the result of , on the other hand , can have any values. A function has left inverse if and only if it is injective (left one-to-one).
An injective function can have several left inverses. This is exactly the case when the function is not surjective and the domain has more than one element.
Left inverses often appear as 'inverses' of embeddings .
For example, consider the number of clubs that have a team in the men's first division in the 2018/19 season . be the amount of municipalities in Germany. The function assigns a club to the municipality in which its stadium is located. Since no two Bundesliga teams come from the same city in the season under review, this function is injective. Since there are also municipalities without a Bundesliga stadium, it is not surjective. So there are several left inverses too . A left inverse that is easy to form is the function that assigns each municipality that owns a Bundesliga stadium to the associated club and all other municipalities to FC Bayern Munich . A more useful example in practice would be the function that each municipality assigns to 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 (linear distance, shortest distance by car, ...).
As a numerical example, let us embed in . Then every rounding function (to 0 places after the comma), for example the Gaussian bracket , can be used as a left inverse. But the function on , which assigns every integer to itself and to all other numbers, is a left inverse.
That is, it fulfills the function
every archetype element can therefore be from below .
If a function has a right inverse, it must be surjective (right total).
Conversely, it seems obvious that the existence of a right inverse follows from the surjectivity of . You can find one or even several original elements for each of them 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 original image elements is actually to be used. Such a simultaneous decision cannot always be made constructively. The axiom of choice (in a suitable formulation) states that a right inverse still 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 favor of the positive solution. In such cases the axiom of choice is not required.
The function is obviously right inverse of if and only if is left inverse of . From this it follows immediately that right inverses are always injective and left inverses are always surjective.
A surjective function has several right inverses if and only if it is not injective.
Right inverse often occurs as functions that determine representatives of a set.
For example, be a function that assigns its genus to each species . As the right inverse , one then chooses a function that names a typical species for each genus. Political representation provides many examples. This could be the nationality of a person, the head of state of a state.
The Hilbert curve continuously maps the unit interval (hence the term curve ) onto the unit square. In practice, however, the Hilbert index is often required, namely a linearization of two-dimensional data (a reversal of the Hilbert curve). For this one takes one of the right inverse of the Hilbert curve, of which there are several - because the Hilbert curve can not be bijective as a continuous mapping between two spaces of different dimensions according to the theorem of the invariance of the dimension .
Left and right inverses of morphisms
If the sets and carry an additional mathematical structure and if an injective or surjective function is compatible with these structures, the question arises whether it is possible to choose the left or right inverse in such a way that the also with the Structures is compatible. This is not the case for many of the structures studied in mathematics. However, if an injective or surjective linear mapping is used, the left or right inverse can also be selected as a linear mapping.
- Functions in which the domain and target set coincide are often of particular interest. For a set , the set of functions of forms a monoid in itself with the composition as a link . The terms of invertibility as well as the left and right inverse that were introduced here then agree with the corresponding terms from algebra.
- The concept of the inverse function in this case is identical to the concept of the inverse element .
- In the general context, the concept of the invertibility of functions is often left out because it corresponds to the concept of bijectivity.
- The above considerations have assumed that and are non-empty. If it is empty, then there is only a function of to if it is also empty.
- This is then the empty function that is bijective and involutor.
- If , but not , is empty, there is exactly one function of nach , which is also empty. This function is injective but not surjective. It has neither left nor right inverse, since there are no functions from after at all .
- There are different approaches to introducing the concept of function in mathematics. The term surjectivity used in this article assumes that the target set is part of the identity of the function. If you take another function concept as a basis, you have to adapt some of the explanations accordingly.
- Most of the statements in this article also apply to functions between classes .
- Helmut Sieber and Leopold Huber: Mathematical terms and formulas for secondary levels I and II in high schools, Ernst Klett Verlag