To determine which of these relations is a function, we need to ensure that each input (first element in each ordered pair) is associated with only one output (second element). In other words, for a relation to be a function, no two ordered pairs should have the same first element but different second elements.

Let's analyze each option:

• Option A: $(1, 0), (3, 0), (1, 1), (3, 1), (1, 3)$

  • Here, the input $1$ appears multiple times with different outputs ($0$, $1$, and $3$), so this is not a function.

• Option B: $(1, 1), (2, 2), (3, 3), (4, 4), (5, 8)$

  • Each input has a unique output. No input is repeated with a different output, so this is a function.

• Option C: $(2, 7), (6, 5), (4, 4), (3, 3), (2, 1)$

  • The input $2$ appears twice with different outputs ($7$ and $1$), so this is not a function.

• Option D: $(9, -3), (9, 3), (4, -2), (4, 2), (0, 0)$

  • The input $9$ has two different outputs ($-3$ and $3$), and the input $4$ also has two different outputs ($-2$ and $2$), so this is not a function.

Conclusion:

The correct answer is B, as it is the only option that satisfies the definition of a function.

Would you like further details, or do you have any questions?

Here are some follow-up questions to deepen understanding:

1. What defines a relation as a function in mathematical terms?
2. How would you determine if a set of points represents a one-to-one function?
3. Can a function have the same output for different inputs? Why or why not?
4. What is the difference between a relation and a function?
5. How would you represent these points visually on a graph to identify if they form a function?

Tip: A quick way to check if a relation is a function is by looking for repeated inputs with different outputs. If they exist, it’s not a function.