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The table you uploaded shows two sections:

1. **"ursprüngliche Tabelle" (original table)**: This table gives values for \( f(x) \), where \( x \) values are 1, 3, 5, and 7, and the corresponding \( f(x) \) (or \( y \)) values are 33.8, 37.4, 41, and 44.6.
   
2. **"inverse Tabelle" (inverse table)**: This table is supposed to represent \( f^{-1}(x) \), the inverse of the original function \( f(x) \). The table is currently empty.

To fill the inverse table, you simply swap the roles of \( x \) and \( y \). That means the values that were in the column for \( f(x) \) (which are 33.8, 37.4, 41, and 44.6) now become the \( x \)-values, and the original \( x \)-values (1, 3, 5, and 7) become the \( f^{-1}(x) \)-values.

Here is how the inverse table should look:

| \( x \)   | \( f^{-1}(x) \) |
|-----------|-----------------|
| 33.8      | 1               |
| 37.4      | 3               |
| 41        | 5               |
| 44.6      | 7               |

Would you like further details or have any questions?

Here are 5 related questions to consider:
1. How can you verify if a function has an inverse function?
2. What are the properties of a one-to-one function that ensures an inverse exists?
3. Can you graph both the original function and its inverse on the same plot?
4. How would you compute the inverse of a function algebraically?
5. What would happen if a function is not one-to-one? Can it still have an inverse?

**Tip**: A function can only have an inverse if it is one-to-one, meaning each \( y \)-value has a unique corresponding \( x \)-value.

The table shows values for f(x) and the inverse function f^(-1)(x). The task is to fill in the inverse table by swapping x and y values from the original table.

Learn how to fill the inverse function table by swapping the x and y values from the original function. This problem demonstrates the concept of inverse functions, suitable for students in Grades 9-11.

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