One occupation is a rule that relates each element of a set A to a single element of a set B. According to this definition, functions must necessarily list all elements of the first set, but not all elements of the second set will be “used”. It is in these two sets that we can find the domain, O counter-domain and the Image of a occupation.
Algebraically, a occupation is defined as follows:
f: A → B
y = f(x)
Where f is the letter chosen to represent a occupation, and y = f(x) is the rule of the function.
The symbol A → B means that the elements of the set A will be evaluated in rule f(x) and will result in an element from the set B. the letter x, in a occupation, represents any element of the set A, so it is called variable: can take any value, as long as this value is one of the elements of A.
Also, x is also independent variable, as it is this variable that determines which element of the set B will be related to the element of set A through the rule y = f(x).
THE variable yes it is dependent
of the variable x, for this reason, is named as the dependent variable. In summary, the variable x represents any element of the set A, and the variable y refers to any element of the set B.What is domain, counter-domain and image?
Given the function y = f (x) that relates the elements of set A to the elements of set B, we can define:
1 - The set A is known as domain. This name is chosen for this set due to the role of its elements in the occupation. Remember that the set A determines the independent variable. Therefore, the elements of the set A have the “domain” over the results of the function, since the results of y obtained depend on the chosen x value.
Example - given the function:
f: N → Z
y = 2x
O set From natural numbers it's the domain, therefore, the numbers that can be related are in the set:
N = {0, 1, 2, 3, 4, 5, 6, 7, …}
2 – Set B is known as counter-domain. This name is chosen because not all elements of set B need to be used for the occupation is valid. Furthermore, this name refers to the dependency that exists between sets A and B.
O counter-domain it's the set where we will find all the numbers that can be related to the elements of the domain through the function f. Taking the previous example again:
f: N → Z
y = 2x
The counterdomain is the set formed by all whole numbers. Note that some whole numbers can never be the result of a multiplication of a natural number by 2, like the number 7. So, although the number 7 belongs to the counter-domain, it cannot be related to any number in the domain.
3 – The subset of counter-domain, formed by all its elements that relate to some element of the domain, is called Image.
So, in the previous role:
f: N → Z
y = 2x
Although the set of all whole numbers is the counter-domain of that occupation, only the even numbers will be the result of some element of the domain applied in the role rule. Therefore, the image set of this function is the set of even numbers.
By Luiz Paulo Moreira
Graduated in Mathematics
Source: Brazil School - https://brasilescola.uol.com.br/o-que-e/matematica/o-que-e-dominio-contradominio-imagem.htm
