The mathematical relationship is the link between the elements of a subset with respect to the product of two sets. A function involves the mathematical operation to determine the value of a dependent variable based on the value of an independent variable. Every function is a relation but not every relation is a function.
Relation Function Definition Subset of ordered pairs that correspond to the Cartesian product of two sets. Mathematical operation to perform with the variable x to get the variable and. Notation
x R. and; x it is related to and.
and=ƒ(x); and is a function of x. Characteristics The sets are not empty. It has a domain and a range. It presents a dependent variable and an independent variable. It has a domain and a range. Examples The seats occupied on a train: the seats on the train are the elements of set A and the people on the train are the elements of set B. The mathematics students of a university: the students of the university are the elements of set A and university majors are the elements of set B. Constant function and=ƒ(x)=c Linear function and=ƒ(x)=ax+b Polynomial function and=ƒ(x)=ax2+bx+c
What is a mathematical relationship?
Any subset C of the Cartesian product A x B is called a binary relation of a set A in a set B or relation between elements of A and B.
That is, if the set A is made up of the elements 1, 2 and 3, and the set B is made up of the elements 4 and 5, the Cartesian product of A x B will be the ordered pairs:
A x B= {(1,4), (2,4), (3, 4), (1,5), (2,5), (3,5)}.
The subset C={(2,4), (3,5)} will be a relation of A and B since it is composed of the ordered pairs (2,4) and (3, 5), the result of the Cartesian product of A x b.
relationship concept
“Let A and B be any two non-empty sets, let A x B be the product set of both, that is: A x B is formed by the ordered pairs (x, y) such that x is the element of A and and it is of B. If in A x B any subset C is defined, a binary relation in A and B is automatically determined in the following way:
x R. and if and only if (x, y) ∈ C
(the notation x R. and means “x it’s related to and“).
We will call the set A starting set and we will call the set B arrival set.
He relationship domain are the elements that make up the starting set, while the ratio range are the elements of the arrival set.
Example of mathematical relationships
Set TO of x male elements of a population and B is the set of and female elements of the same population. A relationship is established when “x is married to and“.
What is a mathematical function?
When we talk about a mathematical function of a set A in a set B, we refer to a rule or mechanism that relates the elements of the set A with an element of the set B.
function concept
“be x and and two real variables, then it is said that y is a function of x if to each value it takes x corresponds to a value of and.”
The independent variable is the x while and is the dependent variable or function:
y=ƒ(x)
The set in which the x is called domain of function (original) and that of the variation of and function range (image).
The set of pairs (x, and) such that and=ƒ(x) is called function graph; If they are represented in some Cartesian axes, a family of points is obtained that is called graph of the function.
function examples
In mathematics we get many examples of functions. Below are examples of flagship functions.
constant function
Graph of the constant function where ƒ (x)=2.
A function is called a constant if the element of set B that corresponds to set A is the same. In this case, all values of x correspond to the same value of y. Thus, the domain is the real numbers while the range is a constant value.
identity function
Graph of the identity function y=ƒ (x)=x.
Let’s suppose x is a variable and that and takes the same value as x. We then have an identity function y=x, where the pairs (x,y) on the graph are (1,1), (2,2), (3,3) and so on.
polynomial function
Graph of the polynomial function ƒ(x)=x2+x-2.
A polynomial function has the form y= anxn+an-1+xn-1+….+a2x2+a1x+a0. The graph above shows the function ƒ(x)=x2+x-2.
Now suppose that the dependent variable and is equal to the independent variable x cubed. We have the function y=x3, whose graph is shown below:
Graph of the function y=ƒ (x)=x3.