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

Options

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

Explanation

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

### Step-by-Step Solution:

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

2. **Set the Denominator to Zero**:
To find the points where the function is undefined, we set the denominator equal to zero:
\[ x^2 - x - 6 = 0 \]

3. **Factor the Quadratic Equation**:
We can factor the quadratic equation:
\[ x^2 - x - 6 = (x - 3)(x + 2) = 0 \]

4. **Find the Roots**:
Setting each factor to zero gives us the roots:
\[ x - 3 = 0 \quad \Rightarrow \quad x = 3 \]
\[ x + 2 = 0 \quad \Rightarrow \quad x = -2 \]

5. **Determine the Domain**:
The function is undefined at \( x = 3 \) and \( x = -2 \). Therefore, the domain of the function is all real numbers except these two points:
\[ \text{Domain of } f(x) = \{ x \in \mathbb{R} \mid x \neq 3 \text{ and } x \neq -2 \} \]

### Final Answer:
The domain of the function \( f(x) \) is \( \mathbb{R} \setminus \{3, -2\} \).

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