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. Since this is a rational function, it is defined for all real numbers except where the denominator is equal to zero.

### Step 1: Identify the denominator
The denominator of the function is:
\[ x^2 - 8x + 12 \]

### Step 2: Set the denominator equal to zero
To find the values that make the denominator zero, we solve the equation:
\[ x^2 - 8x + 12 = 0 \]

### Step 3: Factor the quadratic equation
We can factor the quadratic expression:
\[ x^2 - 8x + 12 = (x - 6)(x - 2) \]

### Step 4: Find the roots
Setting each factor equal to zero gives us the values that make the denominator zero:
\[ x - 6 = 0 \quad \Rightarrow \quad x = 6 \]
\[ x - 2 = 0 \quad \Rightarrow \quad x = 2 \]

### Step 5: Determine the domain
The function \( f(x) \) is undefined at \( x = 6 \) and \( x = 2 \). Therefore, the domain of the function is all real numbers except these two points. In interval notation, the domain can be expressed as:
\[ \text{Domain of } f(x) = \mathbb{R} - \{2, 6\} \]
or
\[ (-\infty, 2) \cup (2, 6) \cup (6, \infty) \]

### Summary of the Domain
Thus, the domain of the function \( f(x) \) is:
\[ \text{Domain} = (-\infty, 2) \cup (2, 6) \cup (6, \infty) \]

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