Question

Input 1, 2, 3, 2, 1
Output 5, 10, 15 20, 24
Definition: A function is a rule that assigns EXACTLY ONE output to each possible input.

This table DOES NOT represent a function.

Change at least one number in the table so that it DOES represent a function.

Explain why you made each change.

Answer 1

To determine how to modify the table so that it represents a function, we first need to analyze the input-output pairs given:

- Input: 1 → Output: 5
- Input: 2 → Output: 10
- Input: 3 → Output: 15
- Input: 2 → Output: 20 (duplicate input)
- Input: 1 → Output: 24 (duplicate input)

In the current set of pairs, the input "2" corresponds to both outputs "10" and "20," and the input "1" corresponds to both "5" and "24." Because a function cannot have more than one output for the same input, the table does not represent a function due to these duplicates.

### Proposed Changes:
1. Change the second output for input `2` from `20` to `12`.
2. Change the second output for input `1` from `24` to `6`.

### Updated Table:
- Input: 1 → Output: 5
- Input: 2 → Output: 10
- Input: 3 → Output: 15
- Input: 2 → Output: 12 (changed from 20)
- Input: 1 → Output: 6 (changed from 24)

### Explanation of Changes:
- **For Input `2`:** By changing the output to `12`, we eliminate the ambiguity with the input `2` having multiple outputs (10 and now 12) instead of 10 and 20. Now, the input `2` only produces a single, unique output.
- **For Input `1`:** By changing the output to `6`, we resolve the same issue, ensuring that input `1` maps to exactly one output (5 and now 6) instead of producing two different outputs (5 and 24).

Now, with unique outputs for each input, the relationship is a valid function as each input corresponds to exactly one output.