https://file.aiquickdraw.com/mathbot/file/1728548193125akmxotd9.jpg

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).

For each relation, decide whether or not it is a function. Analyze Relations 1 to 4 and make a decision based on the mappings between domain and range.

Learn how to determine whether a given relation is a function by analyzing the domain and range. We explore Relations 1 to 4 with detailed explanations for each case. Suitable for students in Grades 9-12.

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