One occupation it is a rule that relates two sets so that each element in the first set has a single representative in the second set. This rule is also known as formation law, and the elements of these sets are called variables.
Domain and Image of a role
The first set of this definition contains numbers that, in a way, dominate your possible results of the function. For this reason, this set is called domain and its elements are called independent variables and, they are usually represented by the letter x.
The second set contains elements that vary according to the variation of domain elements. Therefore, the second set is composed of "images" of the independent variables, since all this set is just the result of each element of the first set evaluated in the law of formation of the occupation. This fact names the second set as Image and its elements like independent variables. These, they are usually represented by the letter y.
To define a function, these two sets must be well defined. To do so, just define the training law and the domain.
Variables are, as in algebraic expressions, numbers represented by letters. The difference lies in the fact that the variable it can take any value within the set to which it belongs, that is, in algebraic expressions, the unknown is an unknown number; in functions, the variable is any number belonging to a numerical set.
Function representations
→ Algebraic representation
The algebraic representation of a occupation is a mathematical formula that relates each element from one set to another. This representation is given by the symbol “f (x)” or the letter “y” with an algebraic expression in the sequence. Below are some examples of function formation laws in their algebraic form.
f (x) = 2x
y = 2x
Note that the two formation laws above refer to the same occupation. If we define the domain of this function as the set of natural numbers, its image will be the set of even numbers. Watch:
f(x) = 2x
f(1) = 2·1 = 2
f(2) = 2·2 = 4
f(3) = 2·3 = 6
…
Substituting x by the natural numbers 1, 2, 3, …, we will always obtain even numbers through the formation law f(x) = 2x. So, 1, 2, 3 … are the elements that make up the domain, and 2, 4, 6 … are the elements that make up the image.
→ Diagram representation
When the function has few elements, it is possible to draw diagrams and link all its elements. In the example below, we will use the same function as the previous example, but with domain restricted to three elements. Watch:
Representation of a function whose domain is D = {1, 2, 3} and the image is I = {2, 4, 6}
degree of a function
The degree of a function is assigned according to the number of variables being multiplied. If the function is given only in one variable (most frequent case), its degree can be evaluated by the highest exponent found among its variables. For example: the function f (x) = 2x has degree 1, since 1 is the largest exponent of a variable present in this function. The function f (x) = x4 – 4x2 has grade 4.
By Luiz Paulo Moreira
Graduated in Mathematics
