https://file.aiquickdraw.com/mathbot/file/1725484480535dppv07g2.jpg

The image displays a table and several questions related to a function \( f(x) \). The table has two columns: \( x \) and \( f(x) \). I'll extract the relevant data from the table and solve the questions below it.

### Table:
\[
\begin{array}{|c|c|}
\hline
x & f(x) \\
\hline
0 & 8 \\
1 & 5 \\
2 & 6 \\
3 & 7 \\
4 & 0 \\
5 & 2 \\
6 & 3 \\
7 & 4 \\
\hline
\end{array}
\]

### Questions:
1. \( f(2) = \) ?
2. If \( f(x) = 9 \), then \( x = \) ?
3. \( f^{-1}(6) = \) ?
4. \( f^{-1}(3) = \) ?

I'll now calculate the answers to each of these questions using the data from the table.Here are the answers based on the table provided:

1. \( f(2) = 6 \)
2. If \( f(x) = 9 \), then \( x = \) No solution (there is no \( x \) for which \( f(x) = 9 \))
3. \( f^{-1}(6) = 2 \) (this means \( f(2) = 6 \))
4. \( f^{-1}(3) = 6 \) (this means \( f(6) = 3 \))

Would you like further explanation on any of these points, or do you have more questions?

### Related Questions:
1. What does the inverse function \( f^{-1}(x) \) represent in general?
2. How do we interpret the table in terms of finding the inverse function?
3. What happens if we encounter a function that is not one-to-one? Can it have an inverse?
4. Can you determine a function that could produce \( f(x) = 9 \)?
5. How do we graphically represent \( f(x) \) and its inverse?

**Tip:** Remember, an inverse function exists only when the original function is one-to-one (each \( y \) value corresponds to exactly one \( x \) value).

Use the table below to fill in the missing values:

1. f(2) = ?
2. If f(x) = 9, then x = ?
3. f⁻¹(6) = ?
4. f⁻¹(3) = ?

Explore how to solve function and inverse function problems using a table of values. Learn how to find f(2), determine x when f(x) equals 9, and identify inverse function values like f⁻¹(6) and f⁻¹(3). Suitable for students in Grades 8-10.

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