- A function requires that each input (x) corresponds to exactly one output (y).
- We test each table for repeated x-values associated with different y-values.
- Table A has unique x-values \((-3, 0, -2, 8)\), so it represents a function.
- Tables B, C, and D are not functions because they each contain at least one x-value that maps to two different y-values.
- The correct table is \(\boxed{A}\).
Explanation
- Understanding the Function Definition
A function is defined as a relation where every input (x) maps to exactly one unique output (y). In the context of sets of ordered pairs or tables, this means that for a table to represent a function, no x-value can be repeated with a different corresponding y-value. If an x-value appears more than once, all instances of that x must be paired with the same y-value. We need to analyze each provided table (A, B, C, D) to see which one satisfies this definition.
- Analyzing Each Table
Let's check each table for repeated x-values:
- Table A: x-values are \(-3, 0, -2, 8\). No x-value is repeated. Since every input is unique, each input maps to exactly one output. Therefore, Table A represents a function.
- Table B: x-values are \(-5, 0, -5, 6\). The x-value \(-5\) appears twice. The corresponding y-values are \(-5\) and \(5\). Since \(-5
eq 5\), this input \(-5\) maps to two different outputs. Therefore, Table B is not a function.
- Table C: x-values are \(-4, -2, -2, 0\). The x-value \(-2\) appears twice. The corresponding y-values are \(2\) and \(4\). Since \(2
eq 4\), this input \(-2\) maps to two different outputs. Therefore, Table C is not a function.
- Table D: x-values are \(-4, 3, 1, -4\). The x-value \(-4\) appears twice. The corresponding y-values are \(2\) and \(0\). Since \(2
eq 0\), this input \(-4\) maps to two different outputs. Therefore, Table D is not a function.
- Conclusion
Based on the analysis, Table A is the only table where each x-value maps to exactly one y-value. All other tables (B, C, and D) contain at least one x-value that maps to two different y-values, violating the definition of a function. Thus, Table A is the correct answer.
Examples
Functions are used in everyday life to describe consistent relationships. For example, consider a vending machine: if you press the button for a specific item (the input), you expect the machine to deliver that specific item (the output) every single time. If you press the button once and it gives you a soda, and then you press the exact same button again and it gives you a candy bar, the vending machine's mechanism is not a "function" because the same input led to two different outcomes. Identifying functions helps us understand and predict outcomes in systems ranging from programming and data analysis to mechanics and economics.