The image you provided contains the following mathematical problem:

Given an inverse $g(x) = \ln(e^x - 2)$, find $f(x)$ such that $f(g(x)) = x$ and $g(f(x)) = x$.

Let's solve this step by step.

Step 1: Express $g(x)$ explicitly.

The function $g(x)$ is given as: $g(x) = \ln(e^x - 2)$

Step 2: Determine the inverse function $f(x)$.

We know that $f(x)$ should be the inverse of $g(x)$, which means: $f(g(x)) = x \quad \text{and} \quad g(f(x)) = x$

Since $g(x)$ is given, we solve $y = g(x)$ for $x$ to find $f(x)$.

1. Start with $y = \ln(e^x - 2)$.
2. Exponentiate both sides to remove the natural logarithm: $e^y = e^x - 2$
3. Solve for $x$: $e^x = e^y + 2$ $x = \ln(e^y + 2)$ Thus, the inverse function $f(x)$ is: $f(x) = \ln(e^x + 2)$

Final Answer:

The function $f(x)$ that satisfies $f(g(x)) = x$ and $g(f(x)) = x$ is: $f(x) = \ln(e^x + 2)$

Would you like more details or have any questions?

Here are five related questions to consider:

1. What is the domain and range of the function $g(x) = \ln(e^x - 2)$?
2. How do you verify that $f(x)$ is indeed the inverse of $g(x)$?
3. What are the properties of logarithmic functions that were used in finding the inverse?
4. How would the inverse change if the function were $g(x) = \ln(e^x - a)$ instead of $g(x) = \ln(e^x - 2)$?
5. Can you generalize the process of finding inverses for other functions involving logarithms and exponentials?

Tip: When finding an inverse function, ensure that the operations you apply can be reversed step by step to return to the original variable.