The problem provides the function $f(x) = 7x + 6$, and asks for the value of the inverse function $f^{-1}(x)$ at $f^{-1}(27)$.

Step-by-step solution:

1. Start with the given function $f(x) = 7x + 6$.
2. To find the inverse $f^{-1}(x)$, swap $f(x)$ with $y$: $y = 7x + 6$
3. Solve for $x$: $y - 6 = 7x$ $x = \frac{y - 6}{7}$
4. So, the inverse function is: $f^{-1}(x) = \frac{x - 6}{7}$
5. Now, calculate $f^{-1}(27)$: $f^{-1}(27) = \frac{27 - 6}{7} = \frac{21}{7} = 3$

Thus, the value of $f^{-1}(27)$ is $\boxed{3}$.

Would you like more details or have any questions?

Related Questions:

1. How do you verify if a function is invertible?
2. What are the conditions for a function to have an inverse?
3. How do you calculate the inverse for a quadratic function?
4. What is the relationship between a function and its inverse graphically?
5. How do you compose a function with its inverse?

Tip: The inverse of a linear function always exists as long as the slope is non-zero.