Let's analyze each relation to determine if it represents a function.

Definitions:

• Function: A relation where each element in the domain is associated with exactly one element in the range. In other words, no domain element is repeated with a different range value.

Relation 1:

• Domain: {c, r, t, v}
• Range: {n, t, j, b}
• Each element in the domain has a unique corresponding value in the range. There are no repeated domain elements with different range values.
  • Conclusion: This is a function.

Relation 2:

• Domain: {-1, -9, 9, 2, -2}
• Range: {desk, pencil, desk, sun, sun}
• Each element in the domain corresponds to a unique value in the range. Even though some range values are repeated (e.g., "desk" and "sun"), the domain elements are not.
  • Conclusion: This is a function.

Relation 3:

• Set: $\{(7, 3), (3, 6), (7, 2), (3, 7)\}$
• Here, we see that the domain element "7" is associated with two different range values (3 and 2), and the domain element "3" is associated with two different range values (6 and 7).
  • Conclusion: This is not a function.

Relation 4:

• Set: $\{(z, -2), (a, -2), (z, -7), (e, 3)\}$
• The domain element "z" is associated with two different range values (-2 and -7), which violates the rule for a function.
  • Conclusion: This is not a function.

Final Answers:

1. Relation 1: Function
2. Relation 2: Function
3. Relation 3: Not a function
4. Relation 4: Not a function

Would you like a more detailed explanation or have any questions?

Here are some related questions:

1. What is the difference between a function and a relation?
2. Can a function have the same range value for different domain values?
3. How would you represent a function using a graph?
4. What is the vertical line test, and how does it determine if a relation is a function?
5. Can a function have more than one output for a single input?

Tip: Remember that in a function, each input (domain element) must map to exactly one output (range element).