We are given the function:

$f(x) = \frac{1}{9}x - 2$

We need to determine its inverse, $f^{-1}(x)$.

Step 1: Replace $f(x)$ with $y$

$y = \frac{1}{9}x - 2$

Step 2: Swap $x$ and $y$

$x = \frac{1}{9}y - 2$

Step 3: Solve for $y$

1. Add 2 to both sides: $x + 2 = \frac{1}{9}y$

2. Multiply both sides by 9 to isolate $y$: $9(x + 2) = y$

3. Expand: $y = 9x + 18$

Step 4: Write the inverse function

$f^{-1}(x) = 9x + 18$

Since none of the provided answer choices match this result exactly, there might be an issue with the choices. Please check if the correct options are given.

Would you like further clarification or a different approach?

Here are 5 related questions to expand your understanding:

1. How do you verify if two functions are inverses of each other?
2. What is the graphical relationship between a function and its inverse?
3. How do you find the inverse of a quadratic function?
4. What are the conditions for a function to have an inverse?
5. How does the inverse of a function affect its domain and range?

Tip: The inverse of a function reflects it across the line $y = x$.