Let's analyze each of the four relations to determine if they are functions.

Relation 1:

• Domain: {desk, sky, pencil}
• Range: {-6, -2, 7, 0, 8}
• Each element in the domain (desk, sky, pencil) maps to a unique element in the range. There are no repeated domain elements with different outputs.
• Conclusion: This is a function.

Relation 2:

• Domain: {7}
• Range: {-2, 7, -6, 0}
• The element "7" in the domain maps to four different values in the range.
• Conclusion: This is not a function because a function must map each domain element to exactly one range element.

Relation 3:

• Set of pairs: {(f, g), (r, x), (g, x), (x, x)}
• There are no repeated domain elements (the first elements of each pair are distinct: f, r, g, x).
• Conclusion: This is a function because each domain element has a unique range element.

Relation 4:

• Set of pairs: {(-3, v), (9, v), (1, v), (9, t)}
• The domain element "9" maps to both "v" and "t", which violates the condition for a function.
• Conclusion: This is not a function.

Summary:

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

Would you like more details on any of the relations, or do you have any other questions?

Here are 5 related questions to expand this topic:

1. What is the formal definition of a function in mathematics?
2. How can we represent functions using graphs or tables?
3. What are one-to-one functions, and how are they different from general functions?
4. Can functions have more than one element in the range mapped from a single domain element?
5. How can we determine if a relation is a function using the vertical line test?

Tip: When dealing with relations, remember that a function must assign exactly one output (range value) to each input (domain element).