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We are given the function \( f(x) = 9x + 4 \), and the task is to find the inverse of this function, denoted by \( f^{-1}(x) \).

### Steps to find the inverse function:
1. Start with the original function:
   \[
   y = 9x + 4
   \]
   where \( y = f(x) \).

2. Swap \( x \) and \( y \) to prepare for solving for the inverse:
   \[
   x = 9y + 4
   \]

3. Solve for \( y \) (which will be the inverse function):
   \[
   x - 4 = 9y
   \]
   \[
   y = \frac{x - 4}{9}
   \]

So, the inverse function is:
\[
f^{-1}(x) = \frac{x - 4}{9}
\]

### Answer:
The correct option is:
\[
f^{-1}(x) = \frac{x - 4}{9}
\]

Do you have any further questions or would you like a more detailed explanation?

### Related Questions:
1. How do you determine if a function has an inverse?
2. Can a function have more than one inverse?
3. What are the steps to verify that two functions are inverses?
4. What is the domain and range of the inverse function?
5. How would you graph both the function and its inverse?

**Tip**: Always check that a function is one-to-one (passes the horizontal line test) before finding its inverse!

For the function f(x) = 9x + 4, find f^{-1}(x).

Learn how to find the inverse of the linear function f(x) = 9x + 4. This step-by-step guide explains the process of swapping variables and solving for y to determine the inverse function f^{-1}(x) = (x - 4)/9. Suitable for students in Grades 9-12.

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