- The range of a relation is the set of all unique y-coordinates.
- From the given ordered pairs \((-8, 6), (6, 1), (7, -5), (-7, -3), (7, -3)\), the y-coordinates are \(\{6, 1, -5, -3, -3\}\).
- Removing duplicates gives \(\{6, 1, -5, -3\}\).
- Sorting the values, the range is \(\boxed{\{-5, -3, 1, 6\}}\).
Explanation
- Defining the Range of a Relation
The range of a relation defined by a set of ordered pairs \((x, y)\) is the set of all unique second coordinates (the \(y\)-values) appearing in those pairs. For the given relation \(S = \{(-8, 6), (6, 1), (7, -5), (-7, -3), (7, -3)\}\), we need to extract the \(y\)-coordinate from each pair and list them as a unique set.
- Identifying the y-coordinates
We list the \(y\)-coordinate from each ordered pair in the set:
- For \((-8, 6)\), the \(y\)-coordinate is \(6\).
- For \((6, 1)\), the \(y\)-coordinate is \(1\).
- For \((7, -5)\), the \(y\)-coordinate is \(-5\).
- For \((-7, -3)\), the \(y\)-coordinate is \(-3\).
- For \((7, -3)\), the \(y\)-coordinate is \(-3\).
The collection of all \(y\)-coordinates found is \(\{6, 1, -5, -3, -3\}\).
- Determining the Final Set
To find the range, we identify the unique values from the collected \(y\)-coordinates. The value \(-3\) appears twice, so we include it only once in our set. The set of unique \(y\)-coordinates is \(\{-5, -3, 1, 6\}\). Ordering these values from least to greatest, we obtain \(\{-5, -3, 1, 6\}\). Comparing this result with the given choices, we confirm it matches the second option. The range is \(\boxed{\{-5, -3, 1, 6\}}\).
Examples
In real life, the concept of a range can be used to describe the set of possible outcomes or results of a process. For example, if you track the daily temperature in a city for a week, the set of unique temperatures recorded constitutes the range of your temperature-tracking relation. Understanding how to extract the range helps in analyzing data trends, such as identifying the minimum and maximum values in a set of observations, which is essential for statistical analysis and reporting.