Does the relation represent a​ function?

A.

The relation in the figure is a function because each element in the domain corresponds to exactly one element in the range.

B.

The relation in the figure is a function because each element in the range corresponds to exactly one element in the domain.

C.

The relation in the figure is not a function because the element

AbelAbel

in the range corresponds to more than one element in the domain.

D.

The relation in the figure is not a function because the element

AmberAmber

in the domain corresponds to more than one element in the range.

To determine whether a relation represents a function, we need to understand the definition of a function:

- A **function** is a relation where each element in the **domain** (the set of inputs) corresponds to exactly **one element in the range** (the set of outputs).

### Let's analyze the provided choices:

1. **Choice A**: 
   - *The relation is a function because each element in the domain corresponds to exactly one element in the range*.
   - This describes the correct condition for a function. If for every input in the domain, there is exactly one output in the range, then it is a function.

2. **Choice B**: 
   - *The relation is a function because each element in the range corresponds to exactly one element in the domain*.
   - This is **incorrect**. A function doesn't require each element in the range to have exactly one corresponding domain element. For instance, two elements from the domain could map to the same element in the range, and it would still be a function.

3. **Choice C**: 
   - *The relation is not a function because the element "Abel" in the range corresponds to more than one element in the domain*.
   - This is **irrelevant** to determining if it's a function. What matters is the behavior of the elements in the **domain**, not in the range.

4. **Choice D**: 
   - *The relation is not a function because the element "Amber" in the domain corresponds to more than one element in the range*.
   - This would indeed indicate that the relation is **not a function**. If any element in the domain maps to more than one element in the range, it violates the definition of a function.

### Conclusion:
- **If each element in the domain maps to only one element in the range, the relation is a function.** 
- The correct answer depends on the behavior of the relation in the figure (whether the domain elements map uniquely to the range). Without seeing the figure, options A or D would be the most relevant to determining if it's a function based on the general conditions for a function.

Would you like me to go into more detail on this or help with something else?

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Here are 5 related questions:
1. How do you determine the domain and range of a function from a relation?
2. Can a relation be a function if multiple domain elements map to the same range element?
3. What is the vertical line test, and how can it help identify functions?
4. How do functions behave when elements in the range have multiple pre-images in the domain?
5. What are examples of relations that are not functions?

**Tip**: Always remember that to qualify as a function, each domain element must map to a unique range element!

Understand how to determine if a relation is a function based on the behavior of the domain and range. This problem explores key mathematical concepts such as functions and relations, making it suitable for students in Grades 9-12.

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