- Set 8: Domain \(\{-4, -3, 0, 1, 2\}\), Range \(\{3, 4, 5, 6\}\), Function \(\boxed{Yes}\) (all \(x\)-values are unique).
- Set 9: Domain \(\{1, 3, 4, 5, 7\}\), Range \(\{0, 7, 4, -2\}\), Function \(\boxed{Yes}\) (each \(x\)-value maps to only one \(y\)-value).
- Set 10: \(\boxed{No \ data \ provided}\).
Explanation
- Analysis of Relations and Functions
A relation is a set of ordered pairs \((x, y)\). The domain of a relation is the set of all unique first components (the \(x\)-values), and the range is the set of all unique second components (the \(y\)-values). A relation is considered a function if each element in the domain maps to exactly one element in the range. This means no two ordered pairs in the set can have the same \(x\)-coordinate while having different \(y\)-coordinates. If an \(x\)-coordinate is repeated, it must be paired with the same \(y\)-coordinate for the relation to remain a function. We will now analyze the provided sets using these criteria.
- Analysis of Set 8
The set is given by \(\{(-4,3), (-3,4), (0,3), (1,5), (2,6)\}\).
- Domain: The set of first components is \(\{-4, -3, 0, 1, 2\}\).
- Range: The set of second components is \(\{3, 4, 5, 6\}\).
- Function: Yes, this set is a function because every \(x\)-coordinate corresponds to exactly one \(y\)-coordinate. Each value in the domain is unique.
- Analysis of Set 9
The set is given by \(\{(1,0), (3,7), (4,4), (5,-2), (7,0)\}\).
- Domain: The set of first components is \(\{1, 3, 4, 5, 7\}\).
- Range: The set of second components is \(\{0, 7, 4, -2\}\).
- Function: Yes, this set is a function because every \(x\)-coordinate corresponds to exactly one \(y\)-coordinate. Even though the \(y\)-value '0' is repeated for two different \(x\)-values, this does not violate the definition of a function. The critical requirement is that no \(x\)-value maps to two different \(y\)-values.
- Analysis of Set 10
No information was provided for set 10. Consequently, it is impossible to determine its domain, range, or function status. For the sake of this analysis, we conclude that the evaluation of this set cannot be completed.
Examples
In real life, identifying functions is crucial for data analysis and predictive modeling. For instance, consider a system where each student (the domain) is assigned exactly one ID number (the range). Because each student has only one ID, this mapping is a function. However, if multiple students were assigned the same ID number, it might be a valid relation, but it would fail the criteria of a unique assignment function. Understanding these mappings helps in organizing databases, scheduling tasks, and managing inventory where unique identification is key.