- A function is one-to-one if each distinct input value maps to a unique output value.
- Examining Table 1, we see repeated output values (11 and 13), so it is not one-to-one.
- Examining Table 2, all output values (12, 7, 9, 13, 5) are unique, confirming it represents a one-to-one function.
- Examining Table 3, we see repeated output values (5 and 2), so it is not one-to-one.
- Therefore, only Table 2 represents a one-to-one function, which is \(\boxed{\text{Table 2}}\).
Explanation
- Defining a One-to-One Function
A function is defined as one-to-one (or injective) if each distinct input value \(x\) in its domain maps to a unique output value \(f(x)\). In other words, a function is one-to-one if no output value \(f(x)\) is repeated for different input values \(x\). To determine which of the three provided tables represents a one-to-one function, we must examine the list of output values for each relation and check for any repeated values.
- Analyzing Table 1
Let's analyze Table 1:
\[ \begin{array}{|c|c|} \ x & f(x) \\ 7.25 & 11 \\ 7.5 & 13 \\ 7.75 & 15 \\ 8 & 13 \\ 8.5 & 11 \\ \end{array} \]
The output values for Table 1 are \(11, 13, 15, 13, 11\). We can see that the value \(11\) is repeated (at \(x = 7.25\) and \(x = 8.5\)) and the value \(13\) is repeated (at \(x = 7.5\) and \(x = 8\)). Since the same output values are produced by different inputs, this relation is not a one-to-one function.
- Analyzing Table 2
Now, let's analyze Table 2:
\[ \begin{array}{|c|c|} \ x & f(x) \\ 9 & 12 \\ 2 & 7 \\ 4 & 9 \\ 1 & 13 \\ 5 & 5 \\ \end{array} \]
The output values for Table 2 are \(12, 7, 9, 13, 5\). Comparing these values, we see that all of them are distinct. Because every distinct input maps to a unique output, Table 2 represents a one-to-one function.
- Analyzing Table 3
Finally, let's analyze Table 3:
\[ \begin{array}{|c|c|} \ x & f(x) \\ -2 & 5 \\ -1 & 2 \\ 0 & 1 \\ 1 & 2 \\ 2 & 5 \\ \end{array} \]
The output values for Table 3 are \(5, 2, 1, 2, 5\). We observe that the value \(5\) is repeated (at \(x = -2\) and \(x = 2\)) and the value \(2\) is repeated (at \(x = -1\) and \(x = 1\)). Since there are repeated output values, this relation is not a one-to-one function.
- Conclusion
By examining the three tables, we have determined that only Table 2 consists of all unique output values. Therefore, Table 2 represents the one-to-one function. The final answer is \(\boxed{\text{Table 2}}\).
Examples
In real-world applications, identifying a one-to-one function is crucial for data integrity. Consider a database storing unique ID numbers (like Social Security Numbers or employee IDs) for individuals. Each ID (the input \(x\)) must map to one and only one person (the output \(f(x)\)). If the same ID was assigned to two different people, the system would fail to differentiate them, leading to errors. By ensuring a function is one-to-one, we guarantee that no two distinct inputs share the same identifier, keeping our records accurate and reliable.