The problem provides the function $f(x) = 3x + 4$ and asks us to find the value of $x$ such that $f^{-1}(x) = 8$.

Solution:

1. Step 1: Find the inverse function $f^{-1}(x)$.

   To find the inverse, start by setting: $y = 3x + 4$ Now, solve for $x$ in terms of $y$.

   $y - 4 = 3x$ $x = \frac{y - 4}{3}$

   So the inverse function is: $f^{-1}(x) = \frac{x - 4}{3}$

2. Step 2: Set the inverse function equal to 8.

   $f^{-1}(x) = 8$ Substituting $x = 8$ into the inverse function:

   $\frac{8 - 4}{3} = \frac{4}{3}$

3. Step 3: Conclusion.

   The value of $x$ for which $f^{-1}(x) = 8$ is: $x = \frac{4}{3}$

Would you like further clarification on any step?

5 Follow-Up Questions:

1. How do you derive the inverse function for other types of equations, such as quadratic functions?
2. What is the graphical interpretation of an inverse function?
3. How do the domain and range of a function change when finding its inverse?
4. What are the conditions for a function to have an inverse?
5. How would the solution change if the function were $f(x) = ax + b$ instead?

Tip: Always check if a function is one-to-one before finding its inverse. This ensures the inverse exists!