Domain of a function
In mathematics we mean by definition set or domain of those subset of a basic amount for which the context requires a well-defined statement is possible. In school mathematics, the definition set is often abbreviated, sometimes that is also written with a double bar.
- 2.1 Example
- 3.1 Examples
- 4.1 Example
Definition of a function
A function is a special relation, which assigns to each element of exactly defined amount of an element of the target amount. The definition set is denoted with. If the function has a name other than such as or, then the domain will be referred accordingly with or.
Of all images of elements of below is called image or set of considered set of all function values it is also called the set of values of. The set of values is a subset of the target set.
The basic amount and the target quantity of a function are essential parts of the definition. But the basic amount and the amount of a target function are not specified Frequently, if the function on the maximum possible set of definitions is meant ( which then usually a subset of the real numbers or complex numbers ).
Two functions with the same functional dependence, but different basic quantities or different target amounts, but are different functions and may have different properties.
Given the picture with the basic set and the target set. Then: is a function of and.
Restriction and continuation of a function
Let be a function and. The function is called the restriction of if for all. ie in this situation extension or continuation of.
The restriction is often written as. This notation is not completely accurate because the amount is not specified; in the interesting cases, but usually selected.
For a function and two given quantities, there are at most a restriction of; it exists if and only if the image set of subset of is.
In contrast to the limitation of an operation continuation is not unique.
Consider the function
Possible continuations on the domain, ie as functions, for example, are both
As well as
Is so far a " nicer " sequel, as is continuous, does not. This does not alter the fact that both functions are proper sequels, as a unique continuation is not received in the function definition itself. Uniqueness arises only from additional demands eg. continuity in this example, or for example in the call for a holomorphic continuation to the complex numbers of a function that is initially defined only on a subset of the real numbers.
Domain of a relation
Under the domain of the relation with
Understood as the projection of, so that subset of elements of the source, which occur as first component in elements:
Consider the relation with
Since the square of real still not negative ( positive), and vice versa for each non-negative real least one real number exists with, for this relation of the domain, the amount of non-negative real numbers.
Domain of definition of a term
The domain of an expression with variables and the corresponding basic sets is the set of all n - tuples, for, for the transitions of the term in meaningful values .
The domain of the term in a variable with the base set is because the fracture is defined only useful for a non-zero value of the denominator.
The domain of the term in two variables with the base set, since the root is defined only useful for non-negative values in the real case.
Domain of equations and inequalities
Are and terms, it is called
And similar expressions are called inequalities. When solving an equation or inequality you are looking for those values from the base portion, for which merges the equation or inequality in a true statement. As a domain is referred to those subset of the base area, for all occurring in the equation or inequality terms are defined sense, ie the average amount of definition of quantity and.
In particular in more complex equations, it can occur that is formed upon release of the initial equation to an equation which also includes solutions that are not included in the domain of the original equation. In such a case, the equation has to be checked after the release, whether the solution obtained values are indeed included in the domain, and optionally some of the values are eliminated.
There are the real solutions of the equation
Sought. Since under the root only non-negative values are allowed, is the domain of the equation.
Squaring the equation gives
Squaring is not equivalent conversion, it is important, but not the transformed equation can therefore contain more solutions than the original equation. Repeated squaring yields
This equation has two solutions. The value is not in the domain of the equation and is therefore not a solution; the value obtained in the initial equation used a true statement and is therefore the only solution of the equation.