To find the inverse function of $f(x) = \sqrt[3]{\ln(5x - 6)}$, follow these steps:

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

$y = \sqrt[3]{\ln(5x - 6)}$

Step 2: Swap $x$ and $y$

$x = \sqrt[3]{\ln(5y - 6)}$

Step 3: Cube both sides to remove the cube root

$x^3 = \ln(5y - 6)$

Step 4: Exponentiate both sides to get rid of the natural logarithm

$e^{x^3} = 5y - 6$

Step 5: Solve for $y$

$5y = e^{x^3} + 6$ $y = \frac{e^{x^3} + 6}{5}$

Thus, the inverse function is: $f^{-1}(x) = \frac{e^{x^3} + 6}{5}$

Would you like more details on how I arrived at this solution or have any questions?

Related questions:

1. How do you find the inverse of more complex functions?
2. What are the steps to differentiate inverse functions?
3. How do natural logarithms and exponentials relate to each other?
4. Can you explain properties of cube roots in detail?
5. What is the significance of the natural logarithm in calculus?

Tip:

When solving for inverse functions, always check the domain and range of both the original and the inverse function to ensure they are valid.