Let's solve for the inverse function $f^{-1}(x)$ for the given function:

$f(x) = 7 + \sqrt{6x - 5}$

Step-by-step solution:

1. Express $y = f(x)$: $y = 7 + \sqrt{6x - 5}$

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

   First, isolate the square root: $y - 7 = \sqrt{6x - 5}$

   Square both sides to eliminate the square root: $(y - 7)^2 = 6x - 5$

   Now, solve for $x$: $6x = (y - 7)^2 + 5$ $x = \frac{(y - 7)^2 + 5}{6}$

3. Replace $y$ with $x$:

   Since we are solving for the inverse function, replace $y$ with $x$: $f^{-1}(x) = \frac{(x - 7)^2 + 5}{6}$

Thus, the inverse function is: $f^{-1}(x) = \frac{(x - 7)^2 + 5}{6}$

This holds for $x \geq 7$.

Do you want more details or have any questions about the process?

Here are 5 related questions to consider:

1. What is the domain and range of $f(x)$ and $f^{-1}(x)$?
2. How does the square root function affect the behavior of the inverse?
3. Can you find the derivative of $f^{-1}(x)$?
4. How do you verify that the two functions $f(x)$ and $f^{-1}(x)$ are indeed inverses?
5. What are the graphical interpretations of inverse functions?

Tip: Always check the domain and range when finding inverse functions to ensure the inverse exists and is correct for the given values.