We are asked to find the inverse of the function $f(x) = 6 + \sqrt{6x - 4}$, specifically $f^{-1}(x)$.

Steps to Find $f^{-1}(x)$:

1. Rewrite the equation for $y = f(x)$: $y = 6 + \sqrt{6x - 4}$

2. Solve for $x$ in terms of $y$:

   • Subtract 6 from both sides: $y - 6 = \sqrt{6x - 4}$
   • Square both sides to remove the square root: $(y - 6)^2 = 6x - 4$
   • Add 4 to both sides: $(y - 6)^2 + 4 = 6x$
   • Divide by 6: $x = \frac{(y - 6)^2 + 4}{6}$

3. Replace $y$ with $x$ to express the inverse function: $f^{-1}(x) = \frac{(x - 6)^2 + 4}{6}$

Thus, the inverse function is:

$f^{-1}(x) = \frac{(x - 6)^2 + 4}{6}, \quad \text{for} \ x \geq 6$

Would you like further explanation on any of these steps?

Here are 5 related questions for further exploration:

1. What are the conditions for a function to have an inverse?
2. How can you verify if a function and its inverse are correct?
3. What role does the domain of $f(x)$ play in determining $f^{-1}(x)$?
4. How would you find the derivative of $f^{-1}(x)$?
5. Can every function be inverted? Why or why not?

Tip: Always check the domain of the original function before solving for the inverse. This ensures that the inverse function's domain is valid and matches the original function's behavior.