- The domain is the set of all unique x-coordinates in a relation.
- The x-coordinates are \(4, 2, -6, -1, 1, -1\).
- The unique values are \(-6, -1, 1, 2, 4\).
- The correct choice is c, which represents the set \(\boxed{\{-6, -1, 1, 2, 4\}}\).
Explanation
- Defining the Domain of a Relation
The domain of a relation is defined as the set of all unique first coordinates (the x-values) present in its ordered pairs. Given the set of ordered pairs \(\{(4,-1), (2,7), (-6,0), (-1,8), (1,5), (-1,-4)\}\), we need to extract these values to determine the domain.
- Extracting the First Coordinates
Let us identify the first coordinate of each ordered pair in the given relation:
- For \((4, -1)\), the x-value is \(4\).
- For \((2, 7)\), the x-value is \(2\).
- For \((-6, 0)\), the x-value is \(-6\).
- For \((-1, 8)\), the x-value is \(-1\).
- For \((1, 5)\), the x-value is \(1\).
- For \((-1, -4)\), the x-value is \(-1\).
The list of first coordinates is: \(4, 2, -6, -1, 1, -1\).
- Forming the Domain Set and Comparing with Options
To form the domain, we identify all distinct values from the list and write them in ascending order. The unique values in our list are \(-6, -1, 1, 2, 4\). Note that the value \(-1\) appears twice, but we only list it once in the domain set. Thus, the domain is the set \(\{-6, -1, 1, 2, 4\}\). Comparing this result with the given choices, we find it matches option c.
Examples
In real life, understanding the domain of a relation is useful when mapping inputs to outputs. For example, if you are tracking the temperature (the range) at different hours of the day (the domain), the domain represents the set of all hours for which you have recorded data. Knowing the domain helps ensure that you only provide inputs that are valid for your system or function, preventing errors and ensuring meaningful results.