To find the inverse function $f^{-1}(x)$ for the function $f(x) = 14x + 6$, follow these steps:

1. Start by replacing $f(x)$ with $y$: $y = 14x + 6$

2. Swap $y$ and $x$ to begin finding the inverse: $x = 14y + 6$

3. Solve for $y$: $x - 6 = 14y$ $y = \frac{x - 6}{14}$

4. Finally, replace $y$ with $f^{-1}(x)$: $f^{-1}(x) = \frac{x - 6}{14}$

So, the inverse function is: $f^{-1}(x) = \frac{x - 6}{14}$

Would you like more details or have any questions about this process?

Here are some related questions you might consider:

1. How do you determine if a function has an inverse?
2. What are the key steps to finding the inverse of a function?
3. How does the graph of a function compare to its inverse?
4. What are the implications of the domain and range when finding an inverse?
5. How can you verify that two functions are inverses of each other?

Tip: Always verify the inverse function by composing it with the original function to see if you get the identity function (i.e., $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$).