Determine whether the relation defines a​ function, and give the domain and range. ​{(9​, 3​), ​(8​, 7​), ​(5​, 1​), ​(2​, 4​)}

To determine whether the given relation defines a function, we need to verify if each input (or first element) is associated with exactly one output (or second element). In other words, for every distinct \( x \)-value, there should be a unique \( y \)-value.

The relation is:  
\[
\{(9, 3), (8, 7), (5, 1), (2, 4)\}
\]

### Step 1: Check if the relation is a function.
- In the ordered pairs, the first elements are \( 9, 8, 5, 2 \), which are all distinct.
- Since each \( x \)-value has exactly one corresponding \( y \)-value, the relation **does define a function**.

### Step 2: Determine the domain.
The domain is the set of all possible \( x \)-values (first elements) in the relation:
\[
\text{Domain} = \{9, 8, 5, 2\}
\]

### Step 3: Determine the range.
The range is the set of all possible \( y \)-values (second elements) in the relation:
\[
\text{Range} = \{3, 7, 1, 4\}
\]

### Conclusion:
- The relation **defines a function**.
- Domain: \( \{9, 8, 5, 2\} \)
- Range: \( \{3, 7, 1, 4\} \)

Would you like further details or have any questions?

Here are 5 related questions to deepen understanding:
1. What conditions must a relation meet to be considered a function?
2. How do we determine the domain and range for more complex relations?
3. Can a relation still be a function if two different \( y \)-values map to the same \( x \)-value?
4. What are the differences between injective, surjective, and bijective functions?
5. How would the answer change if one \( x \)-value was associated with two different \( y \)-values?

**Tip:** Always check the uniqueness of each \( x \)-value's corresponding \( y \)-value to confirm if a relation is a function.

Explore how to determine if a relation defines a function and find its domain and range with the relation {(9, 3), (8, 7), (5, 1), (2, 4)}. This problem focuses on understanding functions, relations, and identifying domain and range. Suitable for students in Grades 8-10.

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