To determine which set of ordered pairs does not represent a function, we need to verify if there are any repeated x-values with different y-values in each set. In a function, each x-value (input) should map to exactly one y-value (output).

Let's analyze each option:

1. ${(-5, -7), (-1, 4), (-3, 4), (5, 2)}$

   • No x-value is repeated with a different y-value. This set represents a function.

2. ${(-8, -7), (4, 7), (4, -9), (5, -6)}$

   • Here, the x-value $4$ is paired with both $7$ and $-9$. This means that this set does not represent a function because an x-value should not map to multiple y-values.

3. ${(-5, -7), (-6, 7), (-1, -9), (9, 7)}$

   • No x-value is repeated with a different y-value. This set represents a function.

4. ${(4, 5), (1, 5), (3, -7), (7, 6)}$

   • No x-value is repeated with a different y-value. This set represents a function.

Conclusion:

The answer is the second option: ${(-8, -7), (4, 7), (4, -9), (5, -6)}$, as it does not represent a function due to the repeated x-value $4$ mapping to different y-values.

Would you like more details on this analysis, or do you have any questions?

Here are five related questions for further exploration:

1. What defines a function in terms of ordered pairs?
2. How can we verify if a set of ordered pairs represents a function?
3. What are real-world examples where a function might have repeated inputs but different outputs?
4. How would you graphically represent a non-function from ordered pairs?
5. How does the concept of a function apply in more complex cases, such as multi-variable functions?

Tip: Remember, for a relation to be a function, each input should have exactly one output.