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The problem asks to find the inverse \( f^{-1}(x) \) for the given function:

\[
f(x) = \frac{x^{\frac{1}{9}}}{9} + 7
\]

### Steps to find the inverse:

1. **Set \( f(x) \) to \( y \)**:
   \[
   y = \frac{x^{\frac{1}{9}}}{9} + 7
   \]
   Now, we want to solve for \( x \) in terms of \( y \).

2. **Isolate the term involving \( x \)**:
   \[
   y - 7 = \frac{x^{\frac{1}{9}}}{9}
   \]

3. **Multiply both sides by 9**:
   \[
   9(y - 7) = x^{\frac{1}{9}}
   \]

4. **Raise both sides to the 9th power to eliminate the fractional exponent**:
   \[
   \left(9(y - 7)\right)^9 = x
   \]

5. **Conclusion: The inverse function \( f^{-1}(x) \) is**:
   \[
   f^{-1}(x) = (9(x - 7))^9
   \]

This matches the first option:

\[
f^{-1}(x) = (9(x - 7))^9
\]

### Would you like more details on the steps or have any questions?

### Related Questions:
1. How do you verify that a function and its inverse are correct?
2. What are the properties of inverse functions?
3. How can you find the domain and range of a function and its inverse?
4. What is the graphical relationship between a function and its inverse?
5. How does the process of finding inverses differ when dealing with different types of functions?

**Tip:** Always check if the function is one-to-one before finding the inverse, since only one-to-one functions have inverses.

For the function f(x) = x^(1/9)/9 + 7, find f^(-1)(x).

Learn how to find the inverse of the function f(x) = x^(1/9)/9 + 7 with a detailed step-by-step solution. This problem covers key algebraic concepts such as inverse functions and exponentiation. Suitable for high school students in Grades 10-12.

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