To determine whether each relation is a function, we need to check whether every input (or $x$-value) corresponds to exactly one output (or $y$-value). Here's the solution for each question in your image:

a) Relation: $(-1, 2), (0, 1), (1, 2), (2, 5)$

• Each $x$-value is unique and corresponds to only one $y$-value.
• Conclusion: This is a function.

b) Relation: $(3, 12), (4, 12), (5, 14), (6, 14)$

• Each $x$-value is unique and maps to exactly one $y$-value. Note that $y$-values can repeat, but that does not affect whether it is a function.
• Conclusion: This is a function.

c) Relation: $(1, 2), (2, 3), (3, 4), (4, 5), (5, 6)$

• Each $x$-value is unique and maps to only one $y$-value.
• Conclusion: This is a function.

d) Table of values:

• The $x$-value $1$ maps to two $y$-values ($-1$ and $1$).
• Similarly, $x = 4$ maps to $-2$ and $2$.
• Conclusion: This is not a function.

e) Table of names and ages:

• Each name corresponds to one age.
• Conclusion: This is a function.

f) Graph of a curve:

• To determine if this is a function, apply the vertical line test: if any vertical line intersects the graph at more than one point, it is not a function.
• The curve fails the vertical line test (e.g., multiple points intersect at some $x$-values).
• Conclusion: This is not a function.

g) Scatterplot:

• Using the vertical line test, no vertical line intersects more than one point.
• Conclusion: This is a function.

Let me know if you'd like detailed explanations for any specific case!

5 Related Questions:

1. How can you identify functions using ordered pairs?
2. What is the importance of the vertical line test for graphs?
3. Why can $y$-values repeat in a function but $x$-values cannot?
4. Can a table with repeated inputs ever be a function?
5. How do real-life situations (like the age example) relate to functions?

Tip:

Always remember: a function assigns exactly one output