To find the inverse of the function $f(x) = -3x - 9$, we need to follow these steps:

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

$y = -3x - 9$

Step 2: Swap $x$ and $y$

$x = -3y - 9$

Step 3: Solve for $y$ (which will be the inverse function)

First, isolate the term involving $y$: $x + 9 = -3y$

Now, solve for $y$: $y = \frac{-(x + 9)}{3}$ $y = -\frac{x + 9}{3}$ $y = -\frac{x}{3} - 3$

Step 4: Write the inverse function

Thus, the inverse function is: $g(x) = -\frac{x}{3} - 3$

Conclusion:

The correct inverse function is $g(x) = -\frac{x}{3} - 3$, which corresponds to the first option:
$g(x) = -\frac{x}{3} - 3$.

Would you like further details or have any questions about this process?

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Here are some related questions:

1. How do you check if two functions are true inverses of each other?
2. How can you find the inverse of a function when it is a more complex expression?
3. What is the significance of the "horizontal line test" in determining if a function has an inverse?
4. Can a function without a one-to-one relationship have an inverse?
5. How do you graph a function and its inverse on the same coordinate plane?

Tip: When finding inverses, always remember to swap $x$ and $y$ and then solve for $y$ to get the inverse.