To find the inverse of the function $f(x) = -3^{x-5} + 3$, follow these steps:

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

$y = -3^{x-5} + 3$

Step 2: Swap $x$ and $y$:

To find the inverse, we swap $x$ and $y$ because the inverse function reverses the roles of inputs and outputs. $x = -3^{y-5} + 3$

Step 3: Solve for $y$:

We now solve for $y$ in terms of $x$.

1. Subtract 3 from both sides: $x - 3 = -3^{y-5}$
2. Multiply both sides by -1: $-(x - 3) = 3^{y-5}$
3. Now, take the logarithm of both sides (logarithm base 3): $\log_3(-(x - 3)) = y - 5$
4. Add 5 to both sides to isolate $y$: $y = \log_3(-(x - 3)) + 5$

Thus, the inverse function is: $f^{-1}(x) = \log_3(-(x - 3)) + 5$

Final Answer:

The inverse of $f(x) = -3^{x-5} + 3$ is: $f^{-1}(x) = \log_3(-(x - 3)) + 5$

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Would you like further clarification or details on any step? Here are some related questions:

1. How do logarithmic functions relate to exponential functions in finding inverses?
2. What is the effect of the negative sign in the function when finding the inverse?
3. How can you confirm if the function and its inverse are correct?
4. What are the general steps for finding the inverse of any function?
5. How do you find the domain and range of a function and its inverse?

Tip: Remember that the domain of the inverse function is the range of the original function, and vice versa!