This question aims to find the relations from the given sets of points that do not fall in the category of functions.

**Relations** and **functions** are two different words having different meanings but both of them talk about the input and output values. The ordered pairs are represented as **(input, output).**

A Function is a type of relationship that gives **only one output value for one input value**. In terms of x and y, a function gives an x-value that is associated with only one y-value. A function always follows this rule. On the other hand, relation shows the relationship between inputs and outputs.

A relation is the **subset** of the **cartesian product**. The relation between the two sets is defined as the collection of ordered pairs. The ordered pairs are created from the objects of each set.

## Expert Answer

The collection of first values of the ordered pairs is called the **domain** while the collection of second values of ordered pairs is called the **range.**

If we consider the following ordered pairs:

\[ A. ( 0 , 8 ) , ( 3 , 8 ) , ( 1 , 6 ) \]

\[ B. ( 4 , 2 ) , ( 6 , 1 ) , ( 8 , 9 ) \]

\[ C. ( 1 , 20 ) , ( 2 , 23 ) , ( 9 , 26 ) \]

\[ D. ( 0 , 3 ) , ( 2 , 3 ) , ( 2 , 0 ) \]

If we consider A then, the domain will be { 0, 1, 3 } and the range is {1, 8}. The given relation gives one output for every input which makes it a function.

\[ B. ( 4 , 2 ) , ( 6 , 1 ) , ( 8 , 9 ) \]

In relation B, the domain will be { 4, 6, 8 } and the range is { 1, 2, 9 }. There is one output for the given relation which means it is a function.

\[ C. ( 1 , 20 ) , ( 2 , 23 ) , ( 9 , 26 ) \]

In relation C, the domain will be {1, 2, 9} and the range is {20, 23, 26}. The given relation qualifies as a function because it has only one output.

## Numerical Solution

**\[ D. ( 0 , 3 ) , ( 2 , 3 ) , ( 2 , 0 ) \]**

In relation B, the domain will be {0, 2} and the range is {0, 3}. This relation is **not a function** because there is not exactly one output for every input. As we can see, **input 2** has **two outputs: 3 and 0.**

## Example

Is the relation ${( -3, 7 ),( -5, 9 ),( -5, 3 )}$ a function?

The domain of this function is {-3, -5} and the range is {3, 7, 9}. This relation is not a function because there is not exactly one output for every input. As we can see, **input -5** has two **outputs: 9 and 3.**

*Image/Mathematical drawings are created in Geogebra.*

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