Let’s analyze each case to determine whether they are functions:

5. Relation: {(3, 7), (4, 7), (5, 7), (6, 7)}

• Analysis: In this set of ordered pairs, each input $x$ has exactly one output $y$. Specifically:
  • $3 \to 7$
  • $4 \to 7$
  • $5 \to 7$
  • $6 \to 7$
• Conclusion: This is a function because no $x$-value is repeated with different $y$-values.

6. Table of values:

• Analysis: In this table:
  • The $x = -7$ is paired with both $y = 1$ and $y = -1$. This means that the same input ($x = -7$) gives two different outputs.
• Conclusion: This is not a function because an input cannot have more than one output.

7. Graph:

• Analysis: This is a vertical set of arrows, with one arrow per $x$-coordinate pointing to various $y$-values. Using the vertical line test:
  • A vertical line passes through more than one point on the graph (e.g., the line at $x = 0$ crosses arrows pointing to different $y$-values).
• Conclusion: This is not a function because the vertical line test fails.

8. Graph:

• Analysis: This graph shows a horizontal line at $y = 2$. Using the vertical line test:
  • A vertical line intersects the graph at only one point for any $x$-value.
• Conclusion: This is a function because the vertical line test is satisfied.

9. Mapping diagram:

• Analysis: In the diagram:
  • $x = 5$ maps to both $y = 7$ and $y = 12$.
• Conclusion: This is not a function because the same input ($x = 5$) has multiple outputs.

Summary of Results:

1. 5: Function
2. 6: Not a function
3. 7: Not a function
4. 8: Function
5. 9: Not a function

Do you want a more detailed explanation or have questions?
Here are 5 related questions to deepen your understanding:

1. What is the formal definition of a function?
2. How can you apply the vertical line test to any graph?
3. Why does having one $x$-value map to multiple $y$-values disqualify a relation as a function?
4. How would you represent a relation that is not a function in a graph or table?
5. Can all mappings be visualized effectively using diagrams or graphs?

Tip: When analyzing whether a relation is a function, always check if each input ($x$) maps to only one output ($y$).