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We are tasked with finding the natural domain of the function:

\[
f(x) = \sqrt{3 - x} + \frac{\ln(x - 1)}{x - 2}
\]

### Step 1: Analyze the square root function
The square root function \( \sqrt{3 - x} \) requires the argument inside the root to be non-negative. So:

\[
3 - x \geq 0 \implies x \leq 3
\]

Thus, the values for \( x \) must be \( x \in (-\infty, 3] \).

### Step 2: Analyze the logarithmic function
The logarithmic function \( \ln(x - 1) \) is only defined when the argument inside the logarithm is positive, so:

\[
x - 1 > 0 \implies x > 1
\]

Thus, for the logarithm, \( x \in (1, \infty) \).

### Step 3: Analyze the denominator
The term \( \frac{\ln(x - 1)}{x - 2} \) includes a denominator \( x - 2 \). To avoid division by zero, we need to exclude the value where \( x = 2 \).

### Step 4: Combine the conditions
We now combine the restrictions from all three parts:

- From the square root: \( x \leq 3 \)
- From the logarithm: \( x > 1 \)
- Exclude \( x = 2 \) to avoid division by zero.

Thus, the domain is:

\[
x \in (1, 2) \cup (2, 3]
\]

This is the valid range of values for \( x \) that satisfy all the conditions.

### Step 5: Conclusion
The correct answer from the given options is:

\[
[1, 2) \cup (2, 3]
\]

Do you have any further questions or would like more details?

### Related Questions:
1. What would the domain be if the logarithmic term were removed?
2. How does the domain change if \( \sqrt{3 - x} \) is replaced with \( \sqrt{x + 3} \)?
3. What happens to the domain if the denominator becomes \( x - 3 \) instead of \( x - 2 \)?
4. How would the range of the function change based on this domain?
5. Could the function still be defined at \( x = 2 \) if \( \ln(x - 1) \) were replaced by another term?

### Tip:
When solving for domains, break the function into parts, analyze restrictions separately, and combine all conditions for a complete solution.

Using interval notation, find the natural domain of the function f(x) = √(3 - x) + ln(x - 1) / (x - 2).

Learn how to find the natural domain of the function f(x) = √(3 - x) + ln(x - 1) / (x - 2) using interval notation. This problem covers key algebraic concepts including logarithmic and square root functions, suitable for students in Grades 10-12.

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