Domain, co-domain and image they are numerical sets used to define the functions. In these sets, there are two types of variable: at independent, which can assume any value belonging to the domain, and dependents, which can assume any value belonging to the counter-domain. To fully understand the concepts of domain, counterdomain and image, it is important to know the concepts of function, variables and sets, which will be discussed below.
Roles
One occupation is a rule that relates each element of the set A to a single element of set B. In other words, a function is a equation which relates numbers that belong to one set to numbers that belong to another according to the above definition.
In the functions, set A is known as domain, and set B is the counter-domain.
Note that two sets and a related rule are needed to define a occupation. Algebraically, we use symbols to represent this definition as follows:
f: A → B
y = f(x)
This symbology means that each element of the set A relates to a single element of the set B through the rule f and that this rule is given by y = f(x). The reading of this symbology is: f from A to B, with y = f (x). Usually, this f(x) is replaced by some equation in occupation of x.
So, given a occupation, for example:
f: N → Z
y = 2x
Realize that the occupation f lists each element of the set of numbersnatural to a single element of the set of numberswhole through the rule y = 2x. Thus, given the elements 1, 2, 3, 4 and 5 of the natural numbers, they will be related to the respective elements of the whole numbers: 2, 4, 6, 8 and 10.
Note that the result y depends on the value chosen for x, so x is called variableindependent and y is called variabledependent.
Domain, co-domain and image
In a function f: A → B, with y = f (x), the domain of that occupation is set A. In other words, the elements that belong to the domain of this function are the same elements that belong to the set THE.
You elements that belong to this set are the possible values of the variableindependent, usually represented by the letter x. For example, consider the following function:
f: N → Z
y = 2x
We know that your domain is composed of all numbersnatural. So the variable x can take any value within that set, but it cannot take any value that does not belong to it.
Note that this occupation get natural numbers from domain and multiply by 2. Therefore, the results obtained when we apply the rule of this function to any number in its domain will be an even number.
O counter-domain is set B, which contains all possible results obtained by applying the function rule to an element of the domain. The counterdomain is a set that must contain all these results. So it's usually a set that contains the domain or is the same as him.
Also note that the counter-domain contains all the values that the variabledependent can assume. This variable is usually represented by the letter y.
In the example given below, note that the elements that belong to the counter-domain of the function are all numberswhole, although not all of them are related to domain elements.
f: N → Z
y = 2x
the image of a occupation is the set of elements of the counter-domain that are related to some element of the domain. In the function above, for example, if x = 2, we have y = 4. The number 4 is called an image of 2 by the function y = 2x. The set of all images is what we call the function image set.
