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.