- A relation is a function if each input value maps to exactly one output value.
- Option 1 is not a function because the input \(-3\) maps to both \(5\) and \(4\).
- Option 3 is not a function because the input \(0\) maps to both \(-1\) and \(0\).
- Option 4 is not a function because the input \(-12\) maps to both \(4\) and \(24\).
- Option 2 represents a valid function as each input corresponds to a single unique output: \(\boxed{\text{Option 2}}\).
Explanation
- Understanding the Definition of a Function
To determine if a set of ordered pairs or a table of values represents a function, we must use the fundamental definition of a function: a relation where every input (the first coordinate, \(x\)) corresponds to exactly one output (the second coordinate, \(y\)). If an input value is associated with two or more different output values, the relation does not qualify as a function.
- Analyzing Option 1
In Option 1, we have the following pairs: \((-5, 10), (-3, 5), (-3, 4), (0, 0), (5, -10)\). We observe that the input \(x = -3\) is associated with two distinct output values: \(y = 5\) and \(y = 4\). Since one input maps to more than one output, Option 1 is not a function.
- Analyzing Option 2
In Option 2, the given set is \(\{(-8, -2), (-4, 1), (0, -2), (2, 3), (4, -4)\}\). Let's list the inputs: \(-8, -4, 0, 2, 4\). Every input appears exactly once, meaning each input is associated with exactly one output value. Therefore, Option 2 satisfies the definition of a function.
- Analyzing Option 3
In Option 3, the table lists inputs: \(-2, -1, 0, 0, 1\). Specifically, the input \(x = 0\) is associated with two distinct output values: \(y = -1\) and \(y = 0\). Because one input maps to more than one output, Option 3 is not a function.
- Analyzing Option 4
In Option 4, the given set is \(\{(-12, 4), (-6, 10), (-4, 15), (-8, 18), (-12, 24)\}\). We notice that the input \(x = -12\) is associated with two distinct output values: \(y = 4\) and \(y = 24\). Because one input maps to more than one output, Option 4 is not a function.
- Conclusion
Based on our analysis, only Option 2 provides a set where every input has a single, unique output. Thus, Option 2 is the only correct answer. \(\boxed{\text{Option 2}}\)
Examples
Imagine a grocery store catalog where every item is assigned a unique product ID. If you type a product ID into the store's computer, it must return exactly one price for that item. If the computer returned two different prices for the same product ID, it would be confusing and inconsistent. This is exactly how functions work in mathematics: every input (product ID) must map to exactly one output (price). If a product ID maps to two different prices, it is not a 'price function'. Understanding this concept helps in creating reliable databases, software, and logical systems.