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### Problem Summary:

Given the function \( f(x) = 2 + \ln x \) and the function \( g: \mathbb{R} \rightarrow \mathbb{R} \), you are asked to:

**(a)** Find the domain of \( f(x) \).

**(b)** Given the equation \( g(f(x)) = -\ln x + 3 \), find the function \( g(x) \).

**(c)** Analyze the monotonicity of \( f(x) \) over the intervals \( (-\infty, 2) \) and \( (2, +\infty) \).

### Solution:

#### (a) Find the domain of \( f(x) = 2 + \ln x \):

The function \( \ln x \) is defined only for \( x > 0 \), so the domain of \( f(x) \) is:

\[
\text{Domain of } f(x): \ (0, +\infty)
\]

#### (b) Find the function \( g(x) \) such that \( g(f(x)) = -\ln x + 3 \):

We know that \( f(x) = 2 + \ln x \). Substitute \( f(x) \) into \( g(f(x)) \):

\[
g(2 + \ln x) = -\ln x + 3
\]

Let \( y = 2 + \ln x \), so \( \ln x = y - 2 \). Substitute back:

\[
g(y) = -(y - 2) + 3 = -y + 2 + 3 = -y + 5
\]

So the function \( g(x) \) is:

\[
g(x) = -x + 5
\]

#### (c) Analyze the monotonicity of \( f(x) = 2 + \ln x \) over the intervals \( (-\infty, 2) \) and \( (2, +\infty) \):

1. **Interval \( (-\infty, 2) \):**

   Since the domain of \( f(x) \) is \( (0, +\infty) \), we must focus only on \( (0, 2) \).

   The derivative \( f'(x) \) is:

   \[
   f'(x) = \frac{d}{dx}[2 + \ln x] = \frac{1}{x}
   \]

   For \( x \in (0, 2) \), \( f'(x) > 0 \), so \( f(x) \) is **increasing** on \( (0, 2) \).

2. **Interval \( (2, +\infty) \):**

   For \( x \in (2, +\infty) \), \( f'(x) > 0 \), so \( f(x) \) is **increasing** on \( (2, +\infty) \).

### Summary of the Results:
1. The domain of \( f(x) \) is \( (0, +\infty) \).
2. The function \( g(x) = -x + 5 \).
3. \( f(x) \) is increasing on both intervals \( (0, 2) \) and \( (2, +\infty) \).

Would you like further details or have any questions?

### Related Questions:
1. What are the properties of the logarithmic function \( \ln(x) \)?
2. How do we find the domain of more complex logarithmic functions?
3. What does it mean for a function to be monotonic over an interval?
4. How do we determine the derivative of composite functions like \( g(f(x)) \)?
5. What is the significance of the function's behavior at the boundaries of its domain?

**Tip:** When dealing with logarithmic functions, always check the domain first to ensure you're working with valid values for \( x \).

Learn how to solve a problem involving logarithmic functions, function composition, and monotonicity. Includes finding the domain of a logarithmic function, determining the composition of functions, and analyzing monotonicity over intervals. Suitable for advanced high school students and those interested in calculus concepts.

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