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.