State the domain and range for the following relation. Then determine whether the relation represents a function. \{(5,-9),(6,-9),(7,-9),(8,-9)\}
Answer (1)
The domain is the set of all first elements: { 5 , 6 , 7 , 8 } .
The range is the set of all second elements: { − 9 } .
Each element in the domain maps to a unique element in the range, so the relation is a function.
The domain is { 5 , 6 , 7 , 8 } , the range is { − 9 } , and the relation represents a function. The domain of the relation is 5 , 6 , 7 , 8 and the range is − 9 . The relation is a function: T r u e .
Explanation
Understanding the Relation The given relation is a set of ordered pairs: { ( 5 , − 9 ) , ( 6 , − 9 ) , ( 7 , − 9 ) , ( 8 , − 9 ) } . We need to identify the domain and range of this relation and determine if it represents a function.
Identifying the Domain The domain of a relation is the set of all first elements (x-values) in the ordered pairs. In this case, the domain is the set { 5 , 6 , 7 , 8 } .
Identifying the Range The range of a relation is the set of all second elements (y-values) in the ordered pairs. In this case, the range is the set { − 9 } . Note that we only list the unique elements, so we don't repeat -9.
Determining if it's a Function A relation is a function if each element in the domain is mapped to exactly one element in the range. In other words, for each x-value, there should be only one corresponding y-value. In our relation, each x-value (5, 6, 7, and 8) is associated with only one y-value (-9). Therefore, this relation represents a function.
Final Answer The domain is { 5 , 6 , 7 , 8 } , the range is { − 9 } , and the relation represents a function.
Examples
Consider a vending machine where each button (domain) corresponds to a specific snack (range). If pressing button 5 always gives you snack -9, button 6 also gives snack -9, and so on, then this vending machine represents a function because each button gives only one specific snack. Understanding domains, ranges, and functions helps in analyzing various real-world relationships, such as input-output systems, data mapping, and cause-effect scenarios.
