Find the domain of the function `f(x) =(x^2+2x+1)/(x^2-8x+12)`

Options

A. R
B. R - {4}
C. R - {2,6}
D. R - {2}

Explanation

To find the domain of the function \( f(x) = \frac{x^2 + 2x + 1}{x^2 - 8x + 12} \), we need to determine the values of \( x \) for which the function is defined. The function is undefined when the denominator equals zero.

### Step-by-Step Solution:

1. **Identify the denominator**:
The denominator of the function is \( x^2 - 8x + 12 \).

2. **Set the denominator to zero**:
We need to find the values of \( x \) that make the denominator zero:
\[ x^2 - 8x + 12 = 0 \]

3. **Factor the quadratic equation**:
To solve the equation, we can factor it:
\[ (x - 2)(x - 6) = 0 \]

4. **Find the roots**:
Setting each factor to zero gives us:
\[ x - 2 = 0 \quad \Rightarrow \quad x = 2 \]
\[ x - 6 = 0 \quad \Rightarrow \quad x = 6 \]

5. **Determine where the function is defined**:
The function \( f(x) \) is undefined at \( x = 2 \) and \( x = 6 \). Therefore, these values must be excluded from the domain.

6. **Write the domain in interval notation**:
The domain of \( f(x) \) can be expressed in interval notation as:
\[ (-\infty, 2) \cup (2, 6) \cup (6, \infty) \]

### Final Domain:
The domain of the function \( f(x) \) is:
\[ \boxed{(-\infty, 2) \cup (2, 6) \cup (6, \infty)} \]