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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.

Givet den bijektive funktion: f(x) = e^{3x} + 6. Angiv værdien for den inverse funktion: f^{-1}(10).

Learn how to solve for the inverse of the exponential function f(x) = e^{3x} + 6. The problem covers the application of logarithms and inverse functions, providing a detailed step-by-step explanation for f^{-1}(10). Suitable for advanced high school students.

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