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.
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