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To find which point is on the graph of \( f^{-1}(x) \), the inverse function, we need to understand the relationship between the function \( f(x) \) and its inverse \( f^{-1}(x) \).

Given the set of points on the graph of \( f(x) \):
\[
\begin{array}{c|c}
x & y \\
\hline
2 & 3 \\
3 & 5 \\
4 & 7 \\
5 & 9 \\
6 & 11 \\
7 & 13 \\
\end{array}
\]

For the inverse function \( f^{-1}(x) \), the roles of \( x \) and \( y \) are swapped. This means that if \( (a, b) \) is on the graph of \( f(x) \), then \( (b, a) \) will be on the graph of \( f^{-1}(x) \).

Let's check the possible points provided:

- Option A: \((-3, -5)\) — This point doesn't match any reverse pair from the given table.
- Option B: \((3, 5)\) — This point is not the reverse of any given pair but rather already in the function table.
- Option C: \((-5, -3)\) — This point also doesn't match any reverse pair from the table.
- Option D: \((5, 3)\) — This point is the reverse of the pair \( (3, 5) \), which is on the table.

Thus, the correct point that is on the graph of \( f^{-1}(x) \) is **(5, 3)**.

Would you like more details or have any questions?

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Here are 5 related questions to deepen your understanding:

1. How do you determine the inverse of a function algebraically?
2. What are the properties of a function that guarantee its inverse will also be a function?
3. How can you graphically find the inverse of a function?
4. Why is the horizontal line test important when discussing inverses of functions?
5. What happens to the domain and range when you take the inverse of a function?

**Tip:** When finding the inverse of a function, always ensure that the original function is one-to-one (passes the horizontal line test), which guarantees the inverse will be a valid function.

Which point is on the graph of f^{-1}(x), the inverse function, given the following set of points on the graph of f(x): (2, 3), (3, 5), (4, 7), (5, 9), (6, 11), (7, 13)?

This problem explores the concept of inverse functions by determining which point lies on the graph of the inverse function given a set of points from the original function f(x). Suitable for students in Grades 9-12.

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