To determine which ordered pair can be removed to make the relation a function, we need to look for pairs that have the same first element (the input value). A relation is considered a function if each input value corresponds to exactly one output value.
The given relation is:
\[ {(0, 0), (2, 0.5), (4, 1), (3, 1.5), (4, 2), (5, 1.5), (6, 8)} \]
Here, we see the input value \(4\) is associated with two different output values:
- \( (4, 1) \)
- \( (4, 2) \)
Since there are two pairs with the same first element (4), we can remove one of these pairs to ensure that each input is unique.
We can choose to remove either \( (4, 1) \) or \( (4, 2) \).
Example of removal:
If we remove the pair \( (4, 1) \), the updated relation would be:
\[ {(0, 0), (2, 0.5), (4, 2), (3, 1.5), (5, 1.5), (6, 8)} \]
This updated relation now has the input value \(4\) associated with exactly one output value \(2\).
Why would removing this ordered pair make the relation a function?
By removing one of the pairs with the same first element, we ensure that no input value (in this case, \(4\)) is associated with more than one output value. Once this is done, every input in the relation will have a unique output, thus satisfying the definition of a function.
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