Removing which ordered pair from the table would make the relation a function of [tex]$x$[/tex]?
| [tex]$x$[/tex] | [tex]$y$[/tex] |
|---|---|
| 1 | 7 |
| 4 | 7 |
| 2 | 2 |
| 4 | 1 |
| 3 | 9 |
A. (1,7)
B. (2,2)
C. (3,9)
D. (4,1)
Answer (1)
The relation is not a function because the x -value 4 is associated with two y -values: 7 and 1. To make the relation a function, we must remove one of the ordered pairs ( 4 , 7 ) or ( 4 , 1 ) . From the given options, removing ( 4 , 1 ) makes the relation a function. Therefore, the answer is ( 4 , 1 ) .
Explanation
Analyzing the Relation A relation is a function if each input x has only one output y . In the given table, we have the following ordered pairs: ( 1 , 7 ) , ( 4 , 7 ) , ( 2 , 2 ) , ( 4 , 1 ) , and ( 3 , 9 ) . We observe that the x -value 4 is associated with two different y -values, 7 and 1. This means the relation is not a function as it is.
Identifying the Problematic Ordered Pairs To make the relation a function, we need to remove one of the ordered pairs that cause the x -value 4 to have multiple y -values. The ordered pairs containing the x -value 4 are ( 4 , 7 ) and ( 4 , 1 ) . The question asks us to choose from the given options: ( 1 , 7 ) , ( 2 , 2 ) , ( 3 , 9 ) , and ( 4 , 1 ) .
Determining the Ordered Pair to Remove If we remove the ordered pair ( 4 , 1 ) , the remaining ordered pairs are ( 1 , 7 ) , ( 4 , 7 ) , ( 2 , 2 ) , and ( 3 , 9 ) . In this case, each x -value is associated with only one y -value. Specifically, x = 1 is associated with y = 7 , x = 4 is associated with y = 7 , x = 2 is associated with y = 2 , and x = 3 is associated with y = 9 . Therefore, the relation becomes a function.
Final Answer Therefore, removing the ordered pair ( 4 , 1 ) from the table would make the relation a function of x .
Examples
In real life, functions are used to model relationships between different quantities. For example, the price of an item might be a function of its demand. If we have a table of data showing the price of an item at different demand levels, we want to make sure that each demand level corresponds to only one price. If we find that one demand level corresponds to two different prices, we need to correct the data to ensure that the relationship is a function.
