- Recall that a function must assign each domain value to exactly one range value.
- Analyze each set by checking for duplicate x-values paired with different y-values.
- Determine that sets A, B, and E have unique x-values, while C and D have duplicate x-values with different y-outputs.
- Identify that the sets representing functions are A, B, and E.
- The final answer is \(\boxed{A, B, E}\).
Explanation
- Understanding the Definition of a Function
A function is a relation between two sets where every element from the first set, called the domain, is mapped to exactly one element in the second set, called the range. In terms of sets of ordered pairs \((x, y)\), this means that for every distinct x-value present in the set, there can be only one corresponding y-value. If an x-value repeats, it must be paired with the same y-value; if an x-value is paired with two different y-values, the relation cannot be a function. We will examine each of the five given sets (A, B, C, D, and E) to verify which ones satisfy this definition.
- Evaluating the Sets
Now, we evaluate each set individually to determine if they meet the requirement for being a function:
- Set A: \(\{(15, -3), (-6, 9), (-3, 0), (-1, 16), (5, 5)\}\)
The domain values are \(\{15, -6, -3, -1, 5\}\). Since every x-value is unique, no value is mapped to more than one output. Thus, Set A is a function.
- Set B: \(\{(6, 0), (-12, -16), (-6, 10), (20, -7), (16, 9)\}\)
The domain values are \(\{6, -12, -6, 20, 16\}\). Since every x-value is unique, no value is mapped to more than one output. Thus, Set B is a function.
- Set C: \(\{(12, -18), (15, 1), (12, 5), (0, 9), (-5, -17)\}\)
The x-value \(12\) appears twice, paired with \(y = -18\) and \(y = 5\). Since \(12\) is mapped to two different values, Set C is not a function.
- Set D: \(\{(-13, 4), (7, -15), (-13, 9), (6, -12), (-18, 0)\}\)
The x-value \(-13\) appears twice, paired with \(y = 4\) and \(y = 9\). Since \(-13\) is mapped to two different values, Set D is not a function.
- Set E: \(\{(10, 9), (-2, -16), (-6, 7), (5, 8), (8, -16), (-11, 9)\}\)
The domain values are \(\{10, -2, -6, 5, 8, -11\}\). Since every x-value is unique, no value is mapped to more than one output. Thus, Set E is a function.
- Conclusion
Based on the analysis, we have identified that Sets A, B, and E satisfy the definition of a function because each input in these sets corresponds to exactly one output. Sets C and D fail this condition due to the presence of repeating x-values associated with different y-values. Therefore, the three sets that represent functions are A, B, and E. The final answer is \(\boxed{A, B, E}\).
Examples
In real-world applications, functions ensure predictable outputs for specific inputs, which is essential for tasks like database management or automated systems. For instance, in a vending machine, each unique product code (the domain) should correspond to exactly one specific snack item (the range). If a single code could dispense two different snacks, the machine would be unreliable. By checking that our sets map each input to only one output, we verify that they function like a reliable system where every action produces a consistent, expected result.