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# Relations and Functions – Explanation & Examples

**Functions and relations are one the most important topics in Algebra**. On most occasions, many people tend to confuse the meaning of these two terms.

In this article, we will define and elaborate on **how you can identify if a relation is a function**. Before we go deeper, let’s look at a brief history of functions.

The concept of function was brought to light by mathematicians in the 17^{th} century. In 1637, a mathematician and the first modern philosopher, Rene Descartes, talked about many mathematical relationships in his book *Geometry. Still, the* term “function” was officially first used by German mathematician Gottfried Wilhelm Leibniz after about fifty years. He invented a notation y = x to denote a function, dy/dx, to denote a function’s derivative. The notation y = f (x) was introduced by a Swiss mathematician Leonhard Euler in 1734.

Let’s now review some key concepts as used in functions and relations.

**What is a set?**

**A set is a collection of distinct or well-defined members or elements**. In mathematics, members of a set are written within curly braces or brackets {}. Members of assets can be anything such as; numbers, people, or alphabetical letters, etc.

For example,

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

{…, −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; they contain the same members. Consider two sets, A = {1, 2, 3} and B = {3, 1, 2}. Regardless of the members’ position in sets A and B, the two sets are equal because they contain similar members.

**What are ordered-pair numbers?**

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

**What is a domain?**

A domain is a **set of all input 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 output or second values. Output values are ‘y’ values of a function.

**What is a function?**

In mathematics, **a function can be defined as a rule that relates every element in one set**, called the domain, to exactly one element in another set, called the range. For example, y = x + 3 and y = x^{2} – 1 are functions because every x-value produces a different y-value.

**A relation**

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

## Types of Functions

*Functions can be classified in terms of relations as follows:*

- Injective or one-to-one function: The injective function f: P → Q implies that there is a distinct element of Q for each element of P.
- Many to one
**:**The many to one function maps two or more P’s elements to the same element of set Q. - The Surjective or onto function: This is a function for which every element of set Q there is a pre-image in set P
- Bijective function.

*The common functions in algebra include:*

- Linear Function
- Inverse Functions
- Constant Function
- Identity Function
- Absolute Value Function

## How to Determine if a Relation is a Function?

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

- Examine the x or input values.
- Examine also the y or output values.
- If all the input values are different, then the relation becomes a function, and if the values are repeated, the relation is not a function.

**Note:** if there is a repetition of the first members with an associated repetition of the second members, the relation becomes a function.

*Example 1*

Identify the range and domain the relation below:

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

__Solution__

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

⟹ {-2, 4, 6}

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

*Example 2*

Check whether the following relation is a function:

B = {(1, 5), (1, 5), (3, -8), (3, -8), (3, -8)}

__Solution__

B = {(1, 5), (1, 5), (3, -8), (3, -8), (3, -8)}

Though a relation is not classified as a function if there is a repetition of x – values, this problem is a bit tricky because x values are repeated with their corresponding y-values.

*Example 3*

Determine the domain and range of the following function: Z = {(1, 120), (2, 100), (3, 150), (4, 130)}.

__Solution__

Domain of z = {1, 2, 3, 4 and the range is {120, 100, 150, 130}

*Example 4*

Check if the following ordered pairs are functions:

- W= {(1, 2), (2, 3), (3, 4), (4, 5)
- Y = {(1, 6), (2, 5), (1, 9), (4, 3)}

__Solution__

- All the first values in W = {(1, 2), (2, 3), (3, 4), (4, 5)} are not repeated, therefore, this is a function.
- Y = {(1, 6), (2, 5), (1, 9), (4, 3)} is not a function because, the first value 1 has been repeated twice.

*Example 5*

Determine whether the following ordered pairs of numbers are a function.

R = (1,1); (2,2); (3,1); (4,2); (5,1); (6,7)

__Solution__

There is no repetition of x values in the given set of ordered pairs of numbers.

Therefore, R = (1,1); (2,2); (3,1); (4,2); (5,1); (6,7) is a function.

*Practice Questions*

- Check whether the following relation is a function:

a. A = {(-3, -1), (2, 0), (5, 1), (3, -8), (6, -1)}

b. B = {(1, 4), (3, 5), (1, -5), (3, -5), (1, 5)}

c. C = {(5, 0), (0, 5), (8, -8), (-8, 8), (0, 0)}

d. D = {(12, 15), (11, 31), (18, 8), (15, 12), (3, 12)}

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