# Functions and relationships

Functions and relationships are one of the most important topics in algebra. In most cases, many people tend to confuse the meaning of these two terms. A relation is a correspondence of elements between two sets. A function is a relation where each element of a set (A) corresponds to one and only one element of another set (B). All functions have a domain and a contradomain. Today in this blog we want to give you information about ** functions and relationships** that will help you understand the meaning of these two concepts of algebra. So we advise you to keep reading this blog and keep discovering very useful information.

Features | Relations |

A mathematical function is the correspondence or relation of each element of a set A with a single element of the set B. The function is the relationship of an element in one set to a single element in another set. Not every relationship is a function. In a graph if you draw a line that the cut can only touch at one point of it. It is a calculating device. The input is the domain, the calculations made by the device with the input are whether the function and the output would be the counter domain. | A relation is any set of ordered pairs, or any correspondence between sets and a function is one that gives exactly one value to the dependent variable (y) for each value of the independent variable (x) in the domain. A relation between two sets A and B is any subset of the Cartesian product AXB, even the vacuum. It is the correspondence between the domain, the range, so that each element of the domain corresponds to one or more elements of the range. |

Index

**What are Functions?**

The concept of function was brought to light by mathematicians in the 17th century. In 1637, a mathematician and the first modern philosopher, René Descartes, spoke about many mathematical relationships in his book Geometry, but the term “function” was officially used for the first time by the German mathematician Gottfried Wilhelm Leibniz after about fifty years. He invented a notation y = x to denote a function, dy / dx to denote the derivative of a function. The notation y = f (x) was introduced by a Swiss mathematician Leonhard Euler in 1734. In mathematics, a function can be defined as a rule that relates each element of a set, called the domain, to exactly one element of another set, called rank. For example, y = x + 3 and y = x2 – 1 are functions because each value of x produces a different value of y.

A function is a special type of relationship in which each element in the domain corresponds to exactly one element in the range. All of this means that a function is an equation in which each value of x is different and does not repeat itself. This means that a function maps the inputs to the outputs and each input produces only one output.

*And we can decide if a relation is a function in two ways:*

Inspect each item in the domain and verify that it only appears once.

Vertical Line Test – This is an easy way to determine if a graph is a function. If we draw vertical lines through a given graph and each line only intersects the graph once, then it is a function. But if each vertical line intersects the graph at more than one point, then the graph is not a function.

**What is a set?**

A set is a collection of distinct or well-defined members or elements. In mathematics, the members of a set are enclosed in braces or brackets {}. The asset members of can be anything like; numbers, people or alphabetic letters, etc.

For instance,

{a, b, c,…, x, y, z} is a set of letters of the alphabet.

{…, −4, −2, 0, 2, 4,…} is a set of even numbers.

{2, 3, 5, 7, 11, 13, 17, …} is a set of prime numbers

Two sets are said to be equal and contain the same members. Consider two sets, A = {1, 2, 3} and B = {3, 1, 2}. Regardless of the position of the members in the set A and B, the two sets are equal because they contain similar members.

**What are the ordered pair numbers?**

These are numbers that go hand in hand. The ordered pair numbers are represented in parentheses and separated by a comma. For example, (6, 8) is an ordered pair number in which the numbers 6 and 8 are the first and second elements respectively.

**What is a domain?**

A domain is a set of all the inputs or first values of a function. Input values are generally “x” values of a function.

**What is a range?**

The range of a function is a collection of all the output values or seconds. The output values are “y” values of a function.

**What are relationships?**

A relation is any set of ordered pair numbers. In other words, we can define a relationship as a group of ordered pairs.

*We can check if a relation is a function either graphically or by following the steps below:*

- Examine the input xo values.
- Also examine the output values and or.

If all the input values are different, then the relation becomes a function, and if the values repeat, the relation is not a function.

Note: If there is repetition of the first members with an associated repetition of the second members, then the relationship becomes a function.

**Example 1**

Identify the range and domain of the relationship below:

{(-2, 3), {4, 5), (6, -5), (-2, 3)}

Solution:

Since the values of x are the domain, the answer is therefore

⟹ {-2, 4, 6}

The range is {-5, 3, 5}.

**What types of functions are there?**

Functions can be classified in terms of relationships as follows:

** Injective or one-to-one** function: The injective function f: P → Q implies that, for each element of P there is a different element of Q.

** Many-to-one** : the many-to-one function assigns two or more elements of P to the same element of the set Q.

** The surjective function or on** : this is a function for which each element of the set Q exists a previous image in the set P

** Bijective function:** A bijective function is a function f that is both surjective and injective. That is, if every element of the final set Y has at least one element of the initial set X to which it corresponds, and all the elements of the initial set X have a single image in the final set Y.

**Common functions in algebra include:**

- Lineal funtion
- Inverse functions
- Constant function
- Identity function
- Absolute value function

I hope you liked all the information that we give you in this blog about functions and relationships in algebra …