- The range of a relation is the set of all unique second coordinates (y-values) in the given ordered pairs.
- By listing the y-coordinates from the pairs \((-8, 6), (6, 1), (7, -5), (-7, -3), (7, -3)\), we get \(\{6, 1, -5, -3, -3\}\).
- Removing the duplicate \(-3\) leaves us with the unique values \(\{6, 1, -5, -3\}\).
- Sorting these values gives the range: \(\boxed{\{-5, -3, 1, 6\}}\).
Explanation
- Understanding the Concept of Range
The given relation is defined by the set of ordered pairs: \((-8, 6), (6, 1), (7, -5), (-7, -3), (7, -3)\). An ordered pair is written in the form \((x, y)\), where \(x\) is the input value (the domain) and \(y\) is the output value (the range). The range of a relation is the set consisting of all unique output values (y-coordinates) present in the ordered pairs.
- Extracting the y-coordinates
To find the range, we identify the second coordinate from each ordered pair:
- From \((-8, 6)\), the y-coordinate is \(6\).
- From \((6, 1)\), the y-coordinate is \(1\).
- From \((7, -5)\), the y-coordinate is \(-5\).
- From \((-7, -3)\), the y-coordinate is \(-3\).
- From \((7, -3)\), the y-coordinate is \(-3\).
The collection of all y-coordinates is \(\{6, 1, -5, -3, -3\}\).
- Identifying Unique Values and Ordering the Range
Now, we identify the unique values within this collection by removing duplicates. The value \(-3\) appears twice, so we include it only once in our set. The set of unique y-coordinates is \(\{6, 1, -5, -3\}\). To present this set in standard form, we order the elements from smallest to largest: \(\{-5, -3, 1, 6\}\).
- Conclusion and Final Answer
Comparing our result \(\{-5, -3, 1, 6\}\) with the provided options, we see that it matches the first option. Thus, the range of the relation is \(\{-5, -3, 1, 6\}\).
Examples
In real-life data management, understanding the range is equivalent to identifying all possible outcomes of a system. For example, if you are tracking the temperature readings in a sensor network throughout the day, the range would be the set of all unique temperature values recorded, which helps in analyzing the variability and climate extremes of a particular location. Determining the range from a set of inputs and outputs allows analysts to define the boundaries within which a process operates.