The given function is $f(x) = e^{3x} + 6$ with the codomain $\mathbb{R}_{\geq 6}$. The task is to find the inverse value for $f^{-1}(10)$.

Step-by-step solution:

1. Start with the equation $f(x) = e^{3x} + 6$.
2. Set $f(x) = 10$, as we are looking for $f^{-1}(10)$: $e^{3x} + 6 = 10$
3. Subtract 6 from both sides: $e^{3x} = 4$
4. Take the natural logarithm (ln) on both sides to isolate $3x$: $3x = \ln(4)$
5. Solve for $x$: $x = \frac{\ln(4)}{3}$

Thus, $f^{-1}(10) = \frac{\ln(4)}{3}$.

Would you like a more detailed explanation or have any questions?

Related questions:

1. How do you find the inverse of a general exponential function?
2. Can you explain the properties of logarithms used in solving for the inverse?
3. What are the applications of inverse functions in real-world scenarios?
4. How does the range of the function affect the determination of the inverse?
5. What happens if the codomain of the function changes?

Tip:

When solving for an inverse function, always ensure that the original function is bijective (both injective and surjective), meaning it has a unique inverse over the specified domain and range.