Let  f (x) = 3 + x + ln(x − 2),  x > 2.

(a)	Find f −1(6)
(b)	Find f −1( f (16))

Given the function \( f(x) = 3 + x + \ln(x - 2) \), where \( x > 2 \), let's go through the two parts of the question.

### Part (a) Find \( f^{-1}(6) \)

To find \( f^{-1}(6) \), we want to determine the value of \( x \) such that \( f(x) = 6 \).

1. Start with the equation:
   \[
   3 + x + \ln(x - 2) = 6
   \]
2. Subtract 3 from both sides:
   \[
   x + \ln(x - 2) = 3
   \]
3. We need to solve this equation for \( x \). This is a transcendental equation, meaning it involves both algebraic and logarithmic terms, and might require numerical methods.

### Part (b) Find \( f^{-1}(f(16)) \)

We know that \( f^{-1}(f(16)) = 16 \), because the inverse function applied to the function itself returns the original input value.

Thus:
\[
f^{-1}(f(16)) = 16
\]

Now, let's proceed by solving \( x + \ln(x - 2) = 3 \) to get \( f^{-1}(6) \). I'll calculate this next.The solution to \( f^{-1}(6) \) is \( x = 3 \).

### Final answers:
(a) \( f^{-1}(6) = 3 \)

(b) \( f^{-1}(f(16)) = 16 \)

Would you like further details on the process, or do you have any questions?

Here are five follow-up questions:
1. How do we find the inverse of a logarithmic function in general?
2. Why does \( f^{-1}(f(x)) = x \) hold for any value of \( x \)?
3. What are transcendental equations, and why are they hard to solve algebraically?
4. What is the domain of \( f^{-1}(x) \) in this problem?
5. How do numerical methods like \( fsolve \) work?

**Tip:** When solving transcendental equations involving logarithms, a good initial estimate helps numerical solvers converge faster.

Let f (x) = 3 + x + ln(x − 2), x > 2.
(a) Find f^{-1}(6)
(b) Find f^{-1}( f (16))

Learn how to find the inverse of the logarithmic function f(x) = 3 + x + ln(x − 2) by solving for f^{-1}(6) and f^{-1}(f(16)). This problem involves transcendental equations and inverse function properties, suitable for advanced high school or undergraduate math students.

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