To ensure that the table represents a function, we need to ensure that each input corresponds to exactly one output. Here's the original input-output mapping:
Input: 1 → 5
Input: 2 → 10
Input: 3 → 15
Input: 2 → 20 (Input 2 mapped to two different outputs)
Input: 1 → 24 (Input 1 mapped to two different outputs)
Since both inputs '2' and '1' map to two different outputs, this violates the definition of a function.
To correct this, we can make the following changes:
- Change the output for one of the inputs that currently maps to multiple outputs.
- For example, we can change the second occurrence of the input '2' (which gives an output of '20') to a distinct output, such as '11'.
Here’s the new mapping:
- Input: 1 → 5
- Input: 2 → 10
- Input: 3 → 15
- Input: 2 → 11 (This mapping is now consistent with input '2' → 10)
- Input: 1 → 24 (This mapping still violates the function definition giving a different output for the same input)
To resolve that, change the last occurrence of '1' output '24' to something else, say '6'.
Thus, the updated mappings would be:
Input: 1 → 5
Input: 2 → 10
Input: 3 → 15
Input: 2 → 11
Input: 1 → 6
If we change the conflicts appropriately and avoid them, one final correct mapping could look something like this:
- Input: 1 → 5
- Input: 2 → 10
- Input: 3 → 15
- Input: 4 → 20
- Input: 5 → 24
Now, here's why the changes work:
- Five distinct inputs (1, 2, 3, 4, 5) each have a unique output (5, 10, 15, 20, 24), fulfilling the definition of a function.
- No input has more than one output, maintaining the rule of exactly one output per input.
The new mapping represents a valid function as every input links to precisely one unique output.