1. For what values of k is the set of order pairs {(2, 4), (k, 6), (4, k)} a one-to-one function

To determine the values of \( k \) for which the set of ordered pairs \(\{(2, 4), (k, 6), (4, k)\}\) forms a one-to-one function, we need to ensure that each input (or domain value) corresponds to exactly one output (or range value).

### Steps:
1. **Identify the input-output pairs**:
   - \( (2, 4) \): Input \( x = 2 \), Output \( y = 4 \)
   - \( (k, 6) \): Input \( x = k \), Output \( y = 6 \)
   - \( (4, k) \): Input \( x = 4 \), Output \( y = k \)

2. **Check for distinct inputs**:
   For the function to be one-to-one, the input values \( 2 \), \( k \), and \( 4 \) must be distinct.
   - If \( k = 2 \), then the pair \( (k, 6) = (2, 6) \), which conflicts with the first pair \( (2, 4) \) because a single input \( 2 \) would correspond to two different outputs, 4 and 6. Therefore, \( k \) cannot be 2.
   - If \( k = 4 \), then the pair \( (4, k) = (4, 4) \) conflicts with the given pair \( (4, k) = (4, k) \), implying the output of \( 4 \) would correspond to two different inputs \( 4 \) and \( k \). However, if \( k = 4 \), this would also create a problem as the input 4 would correspond to both 4 and the value of \( k \).

   Hence, \( k \) must be different from both 2 and 4.

3. **Determine the distinct outputs**:
   - We already have the outputs \( 4 \) and \( 6 \). For \( k \) to be valid, the output \( y = k \) must be distinct from the other outputs (i.e., \( k \neq 4 \) and \( k \neq 6 \)).

### Conclusion:
For the function to be one-to-one:
- \( k \) must be different from 2 and 4 (from the input condition).
- \( k \) must also be different from 4 and 6 (from the output condition).

Thus, \( k \) must satisfy the condition:
\[
k \neq 2, \quad k \neq 4, \quad \text{and} \quad k \neq 6
\]
So, \( k \) can be any value other than 2, 4, or 6.

### Summary:
The set of ordered pairs \(\{(2, 4), (k, 6), (4, k)\}\) forms a one-to-one function if \( k \) is any real number except \( k \neq 2, 4, 6 \).

Would you like more details or have any further questions?

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**Related Questions:**
1. What is a one-to-one function, and how can it be identified?
2. How does the vertical line test relate to determining if a function is one-to-one?
3. What happens when a function has repeating input values?
4. How do injective (one-to-one) functions differ from surjective (onto) functions?
5. How can we determine the range of a function given specific conditions on its input?

**Tip:** Remember that in a one-to-one function, no two distinct inputs can produce the same output, and each input must produce a unique output.

Explore how to identify one-to-one functions using an example problem. Understand the conditions under which a set of ordered pairs forms a one-to-one function. Suitable for high school students studying functions and relations.

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