Determine if the following relation is a function and explain why.

(3, 2),  (-2, 4),  (-4, 2),   (5, 1)



Yes, each domain (input) is only used once

No, there is an element of the domain (input) that is used more than once

Yes, each range (output) is only used once

No, there is an element of the range (output) that is used more than once

To determine if a given relation is a function, we need to check if each input (domain element) is associated with exactly one output (range element).

### Analyzing the Given Relation:

The relation provided is a set of ordered pairs:  
\[(3, 2), \ (-2, 4), \ (-4, 2), \ (5, 1)\]

- **Domain (inputs):** These are the first elements in each pair: \(3, -2, -4, 5\)
- **Range (outputs):** These are the second elements in each pair: \(2, 4, 2, 1\)

### Criteria for a Function:

A relation is a function if **each element of the domain (input)** is mapped to **exactly one element of the range (output)**. 

### Checking the Domain:

- Each element in the domain (inputs \(3, -2, -4, 5\)) appears **only once**.
- There are no repeated inputs with different outputs.

### Checking the Range:

- The range elements (outputs) are \(2, 4, 2, 1\).
- Although the output value \(2\) appears twice in the range, this **does not affect whether the relation is a function**. Functions can have repeated outputs.

### Conclusion:

- **Yes, each domain (input) is only used once**, and each input is associated with exactly one output.
- Therefore, the relation **is a function**.

### Answer:

**Yes, each domain (input) is only used once.** This confirms the relation is a function.

Would you like more details or have any questions?

Here are some related questions you might find interesting:
1. How can we determine if a relation is not a function?
2. Can a function have the same output for different inputs?
3. What are examples of relations that are not functions?
4. How does the concept of a function relate to real-world scenarios?
5. What is the difference between a function and a one-to-one function?

**Tip:** Remember that to check if a relation is a function, focus on whether each input (domain) is paired with a unique output (range). Repeated outputs are acceptable, but repeated inputs with different outputs are not.

Understand how to determine if a relation is a function with clear examples and explanations. Learn the criteria for functions, including the uniqueness of domain elements and the possibility of repeated range elements.

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