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The image you provided is a math exercise that deals with inverse functions, specifically based on a table and the formula of a given function \( f(x) = \left( \frac{x - 9}{5} \right) + 32 \), which converts Celsius to Fahrenheit.

Here’s a breakdown of what needs to be done for each task:

1. **Table:**
   - You need to complete the inverse table using the values from the original table. This means switching the values of \( x \) and \( f(x) \). For example, if \( f(x) = y \), then for the inverse, \( f^{-1}(y) = x \).

   Original Table:
   \[
   \begin{array}{|c|c|}
   \hline
   x & f(x) \\
   \hline
   1 & 33.8 \\
   3 & 37.4 \\
   5 & 41 \\
   7 & 44.6 \\
   \hline
   \end{array}
   \]

   Inverse Table:
   \[
   \begin{array}{|c|c|}
   \hline
   x & f^{-1}(x) \\
   \hline
   33.8 & 1 \\
   37.4 & 3 \\
   41 & 5 \\
   44.6 & 7 \\
   \hline
   \end{array}
   \]

2. **Explain the significance of the function and its inverse:**
   The function \( f(x) \) converts Celsius to Fahrenheit using the formula \( f(x) = \left( \frac{x - 9}{5} \right) + 32 \). The inverse function \( f^{-1}(x) \) would convert Fahrenheit back to Celsius, which means that it undoes the conversion process. In practical terms, if you know the Fahrenheit temperature, you can use \( f^{-1}(x) \) to find the corresponding Celsius temperature.

3. **Calculate the inverse function:**
   You are asked to compute the inverse function of \( f(x) \). Start with the given function:
   \[
   f(x) = \left( \frac{x - 9}{5} \right) + 32
   \]
   To find the inverse:
   - Replace \( f(x) \) with \( y \):
     \[
     y = \left( \frac{x - 9}{5} \right) + 32
     \]
   - Solve for \( x \):
     \[
     y - 32 = \frac{x - 9}{5}
     \]
     \[
     5(y - 32) = x - 9
     \]
     \[
     x = 5(y - 32) + 9
     \]
   So, the inverse function is:
   \[
   f^{-1}(x) = 5(x - 32) + 9
   \]
   This is the formula to convert Fahrenheit back to Celsius.

4. **Check for \( f^{-1}(98) = 37 \):**
   Substituting \( x = 98 \) into the inverse function:
   \[
   f^{-1}(98) = 5(98 - 32) + 9 = 5(66) + 9 = 330 + 9 = 339
   \]
   This doesn't match \( 37 \), so there might be a mistake in the problem setup or interpretation. You might need to recheck the formula or units.

Graphs: 
- The first graph shows the function and its inverse reflected along the line \( y = x \).
- The second graph demonstrates the logarithmic function and its reflection.
- The third graph compares various inverse functions.

Would you like me to further elaborate on any of these points, or provide more details? 

### 5 Related Questions:
1. What is the relationship between a function and its inverse in terms of their graphs?
2. How does the formula for converting Celsius to Fahrenheit work?
3. What is the process of finding the inverse of a more complex function?
4. How do we interpret the graphical reflection of a function and its inverse?
5. Can inverse functions always be found for any function?

### Tip:
When solving for the inverse of a function, always ensure that the function is one-to-one (injective) to guarantee the existence of a well-defined inverse.

The problem deals with inverse functions, specifically converting Celsius to Fahrenheit using a table of values. The user is tasked with completing an inverse function table, explaining the significance of the function, calculating the inverse function, and testing the inverse for a given value.

Learn how to calculate inverse functions by converting Celsius to Fahrenheit with a step-by-step solution. This exercise includes completing a table of values, finding the inverse function formula, and testing the results. Suitable for Grades 9-12 students.

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