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To solve the given problem, we need to determine the domain of the function \( f(x) \), which is:

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

The domain of a function is defined as all real numbers \( x \) for which the function is defined. This requires checking the denominators, as division by zero is undefined.

### Step 1: Analyze the first term
The first fraction is:

\[
\frac{(x - 3)^2}{x^2 - 2x + 1}
\]

We need to set the denominator equal to zero to find the values that make this fraction undefined:

\[
x^2 - 2x + 1 = 0
\]

This can be factored as:

\[
(x - 1)^2 = 0
\]

Thus, \( x = 1 \) makes the first term undefined.

### Step 2: Analyze the second term
The second fraction is:

\[
\frac{x + 2}{(x - 2)^2}
\]

The denominator here is \((x - 2)^2\). Setting this equal to zero gives:

\[
(x - 2)^2 = 0
\]

Thus, \( x = 2 \) makes the second term undefined.

### Step 3: Determine the domain
The function is undefined for \( x = 1 \) and \( x = 2 \). Therefore, the domain is all real numbers except \( x = 1 \) and \( x = 2 \), which is:

\[
R - \{1, 2\}
\]

### Correct answer:
The correct option is **D: \( R - \{1, 2\} \)**.

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Would you like more details on any step?

Here are 5 related questions to deepen your understanding:

1. How do we determine the domain of a rational function in general?
2. Why does dividing by zero make the function undefined?
3. How does factoring quadratic equations help in finding undefined values?
4. What are other situations that might limit the domain of a function?
5. How would the domain change if the numerator also had restrictions?

**Tip**: Always remember to check the entire function, especially for square roots and denominators, to find all domain restrictions.

Determine el dominio de la siguiente función: f(x) = ((x - 3)^2 / (x^2 - 2x + 1)) - ((x + 2) / (x - 2)^2)

Learn how to find the domain of the rational function f(x) = ((x - 3)^2 / (x^2 - 2x + 1)) - ((x + 2) / (x - 2)^2). This solution involves factoring and checking for domain restrictions, ensuring that the function is defined for all real values except where denominators are zero.

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