https://file.aiquickdraw.com/mathbot/file/1723274212595673l468s.jpg

To solve the problem, let's break it down step by step:

### Problem Statement
Given:
\[ f(g(x)) = g(f(x)) = x \]
\[ g(x) = 2 + \ln(x + 1) \]

We need to find \( f(4) \).

### Solution
Since \( f(g(x)) = x \), this implies that \( f(x) \) is the inverse of \( g(x) \). Therefore, \( f(x) = g^{-1}(x) \).

Now, to find \( f(4) \), we need to determine the value of \( x \) such that:
\[ g(x) = 4 \]

Given:
\[ g(x) = 2 + \ln(x + 1) \]

Set \( g(x) = 4 \) and solve for \( x \):
\[
2 + \ln(x + 1) = 4
\]
Subtract 2 from both sides:
\[
\ln(x + 1) = 2
\]
Now, exponentiate both sides to remove the logarithm:
\[
x + 1 = e^2
\]
Subtract 1 from both sides:
\[
x = e^2 - 1
\]

So, \( f(4) = e^2 - 1 \).

### Final Answer
\[ f(4) = e^2 - 1 \]

### Do you want details or have any questions?

Here are 8 related questions:
1. How do you find the inverse of a logarithmic function in general?
2. What are the properties of inverse functions?
3. Why does the equation \( f(g(x)) = x \) imply that \( f(x) \) is the inverse of \( g(x) \)?
4. How do you solve equations involving natural logarithms and exponents?
5. What is the significance of the constant "2" in the function \( g(x) = 2 + \ln(x + 1) \)?
6. Can \( g(x) = 2 + \ln(x + 1) \) have more than one inverse?
7. What is the domain of the function \( g(x) = 2 + \ln(x + 1) \)?
8. How do you graph \( g(x) = 2 + \ln(x + 1) \) and its inverse function?

**Tip:** Understanding how logarithmic and exponential functions are inverses of each other is key to solving problems involving these functions.

Learn how to solve the problem involving the functions f and g where f(g(x)) = g(f(x)) = x. This problem explores inverse functions, logarithmic functions, and exponential functions with a detailed step-by-step solution. Suitable for advanced high school students.

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