We are asked to find the inverse of the function $f(x) = 3^{x-1}$. Let's go through the steps:

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

$y = 3^{x-1}$

Step 2: Swap $x$ and $y$

To find the inverse, we switch $x$ and $y$: $x = 3^{y-1}$

Step 3: Solve for $y$

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

• Apply the logarithm base 3 to both sides: $\log_3(x) = y - 1$

• Solve for $y$: $y = \log_3(x) + 1$

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

Conclusion:

The correct answer is: $\boxed{\log_3(x) + 1}$

Would you like further details or have any questions? Here are five related questions to deepen understanding:

1. How do you calculate the inverse of other exponential functions?
2. What is the relationship between logarithms and exponents?
3. How do you solve logarithmic equations for an unknown variable?
4. How do changes in the base of the exponent affect the inverse function?
5. Can you explain how to derive the inverse of more complex functions involving exponents?

Tip: When solving for inverses, always remember to swap the variables and apply logarithmic or exponential rules as needed!