To determine which set of ordered pairs represents a function, we need to remember that a function cannot have the same input (first element of the ordered pair) map to more than one output (second element of the ordered pair). In other words, no two pairs in the set can have the same first number (x-value) with different second numbers (y-values).

Let's analyze each set of ordered pairs:

1. Set 1: $\{(1, 7), (8, -9), (1, -2), (-6, -1)\}$

   • The first element $1$ appears twice: in $(1, 7)$ and $(1, -2)$, which means the same input is associated with two different outputs.
   • This is not a function.

2. Set 2: $\{(9, 0), (7, 9), (3, -4), (3, -6)\}$

   • The first element $3$ appears twice: in $(3, -4)$ and $(3, -6)$, which again violates the definition of a function.
   • This is not a function.

3. Set 3: $\{(-2, -6), (-5, -6), (9, -2), (7, 3)\}$

   • All first elements $(-2, -5, 9, 7)$ are distinct, so each input maps to exactly one output.
   • This is a function.

4. Set 4: $\{(4, 7), (8, 1), (-6, -1), (-6, -5)\}$

   • The first element $-6$ appears twice: in $(-6, -1)$ and $(-6, -5)$, which violates the function rule.
   • This is not a function.

Conclusion:

• The correct answer is Set 3, $\{(-2, -6), (-5, -6), (9, -2), (7, 3)\}$, as each input corresponds to exactly one output.

Would you like further explanation on any part of this?

Here are five related questions to deepen understanding:

1. What is the definition of a function in terms of ordered pairs?
2. How can you tell if a graph represents a function?
3. Can a function have the same output for different inputs?
4. What is the vertical line test, and how does it relate to functions?
5. How do domain and range relate to functions?

Tip: Always check for repeated first elements (inputs) when identifying functions from a set of ordered pairs.