- Identify the y-coordinates from the given ordered pairs \((-1, 8), (0, 3), (1, -2), (2, -7)\).
- Extract the output values: \(\{8, 3, -2, -7\}\).
- Sort the values: \(\{-7, -2, 3, 8\}\).
- The final range is \(\boxed{\{y: y = -7, -2, 3, 8\}}\).
Explanation
- Analyzing the Given Set of Ordered Pairs
A function is a relation that assigns each element of its domain to exactly one element in its range. A set of ordered pairs \((x, y)\) defines a function where the first coordinate \(x\) belongs to the domain and the second coordinate \(y\) belongs to the range. Given the set of ordered pairs \((-1, 8), (0, 3), (1, -2), (2, -7)\), we need to identify the set of all unique output values (the y-coordinates) to determine the range of the function.
- Extracting and Organizing the Range Values
To find the range, we examine each ordered pair \((x, y)\) and extract the y-coordinate:
- From \((-1, 8)\), the output is \(8\).
- From \((0, 3)\), the output is \(3\).
- From \((1, -2)\), the output is \(-2\).
- From \((2, -7)\), the output is \(-7\).
The set of these output values is \(\{8, 3, -2, -7\}\). Sorting these values in ascending order, we obtain \(\{-7, -2, 3, 8\}\).
- Conclusion and Final Answer
Comparing our set \(\{-7, -2, 3, 8\}\) with the provided options:
- \(\{x: x = -1, 0, 1, 2\}\) represents the domain, not the range.
- \(\{y: y = -7, -2, 3, 8\}\) matches our extracted values.
- \(\{x: x = -7, -2, -1, 0, 1, 2, 3, 8\}\) is a combination of domain and range elements.
- \(\{y: y = -7, -2, -1, 0, 1, 2, 3, 8\}\) is also an incorrect combination.
Thus, the range of the function is \(\{y: y = -7, -2, 3, 8\}\).
Examples
In real life, functions help model relationships between different variables. For example, if you are tracking the temperature at different times, each hour might correspond to a specific recorded temperature. The set of all hours would be your domain, and the set of all observed temperatures would be your range. Identifying these sets is crucial for understanding the behavior of the system, such as finding the minimum or maximum values within a specific data set.