The domain of the function f given by f(x) = `(x^(2)+ 2x+ 1)/(x^(2) - 5x+6)`

Options

A. R - { 3,-2}
B. R - {-3,2}
C. R - {3,-2}
D. R - {3,-2}

Answer

To find the domain of the function \( f(x) = \frac{x^2 + 2x + 1}{x^2 - 5x + 6} \), we need to determine the values of \( x \) for which the function is defined. The function will be undefined wherever the denominator is equal to zero.

### Step-by-Step Solution:

1. **Identify the Denominator**:
The denominator of the function is \( x^2 - 5x + 6 \).

2. **Set the Denominator to Zero**:
To find the values that make the function undefined, we set the denominator equal to zero:
\[ x^2 - 5x + 6 = 0 \]

3. **Factor the Quadratic**:
We can factor the quadratic equation:
\[ (x - 2)(x - 3) = 0 \]

4. **Solve for \( x \)**:
Setting each factor to zero gives us:
\[ x - 2 = 0 \quad \Rightarrow \quad x = 2 \]
\[ x - 3 = 0 \quad \Rightarrow \quad x = 3 \]

5. **Determine the Domain**:
The function is undefined at \( x = 2 \) and \( x = 3 \). Therefore, the domain of the function excludes these values. In interval notation, the domain can be expressed as:
\[ \text{Domain of } f = \mathbb{R} \setminus \{2, 3\} \]
or in interval notation:
\[ (-\infty, 2) \cup (2, 3) \cup (3, \infty) \]

### Final Answer:
The domain of the function \( f \) is \( (-\infty, 2) \cup (2, 3) \cup (3, \infty) \).

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