Let's go through each question:

Question 1

This question asks: "Which relation is NOT a function?"

To identify whether each option is a function, recall that a function should map each input to exactly one output. In other words, each $x$-value should have only one $y$-value.

1. Option A: A mapping diagram. Here:

   • $x = 2$ maps to $y = 6$.
   • $x = 3$ maps to both $y = 2$ and $y = 3$.

   Since $x = 3$ maps to two different $y$-values, Option A is not a function.

2. Option B: A table of values.

   • Each $x$-value (1, 2, and 3) maps to a single $y$-value (6).

   Therefore, Option B is a function.

3. Option C: A set of ordered pairs $(1, 2), (1, 3), (1, 4)$.

   • Here, $x = 1$ maps to multiple $y$-values (2, 3, and 4).

   This violates the rule for a function, so Option C is not a function.

4. Option D: A graph.

   • By using the vertical line test (a line drawn vertically should intersect the graph at only one point), we can see that some vertical lines intersect this graph at more than one point.

   Option D is also not a function.

Answer: Since both Option A, C, and D are not functions, if only one answer is expected, the question may contain an error. However, typically, either A or C would likely be the correct answer if we must choose one.

Question 2

This question asks: "Select all relations that are functions." There are two correct answers.

1. Option A: The function $y = \frac{1}{2} x + \frac{1}{3}$.

   • This is a linear equation, which defines a function where each $x$-value corresponds to exactly one $y$-value.

   Option A is a function.

2. Option B: A table with pairs of $(x, y)$ values.

   • Each $x$-value maps to a single $y$-value in the table.

   Option B is a function.

3. Option C: A graph.

   • By performing the vertical line test, we can see that each vertical line intersects the graph only once.

   Option C is a function.

4. Option D: A set of ordered pairs $(-1, 0), (0, 1), (1, 1), (2, 2)$.

   • Each $x$-value has a unique $y$-value, so this is a function as well.

   Option D is a function.

Answer: Since there are two correct answers, Options A, B, C, and D are all valid functions. However, the question instructs to select only two. Likely correct answers are B and C based on common interpretation patterns, but all technically qualify as functions.

Would you like further details on any of these solutions, or have questions?

Here are five related questions to extend your understanding:

1. How does the vertical line test determine if a graph represents a function?
2. Can a relation with repeating $x$-values ever be a function?
3. Why is it important that each $x$-value in a function has only one $y$-value?
4. What are some common real-world examples of functions and non-functions?
5. How can you identify a function from a set of ordered pairs?

Tip: When examining tables or sets for functions, check each $x$-value for multiple $y$-values to quickly identify if it fails the function requirement.