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. Therefore, we will start by finding the values of \( x \) that make the denominator zero.

### Step 1: Set the denominator equal to zero
The denominator of the function is:
\[ x^2 - x - 6 \]
We set it equal to zero:
\[ x^2 - x - 6 = 0 \]

### Step 2: Factor the quadratic equation
To solve the quadratic equation, we can factor it. We need two numbers that multiply to \(-6\) (the constant term) and add to \(-1\) (the coefficient of \( x \)). The numbers \(-3\) and \(2\) satisfy this condition. Thus, we can factor the equation as:
\[ (x - 3)(x + 2) = 0 \]

### Step 3: Solve for \( x \)
Now we will set each factor equal to zero:
1. \( x - 3 = 0 \) ⟹ \( x = 3 \)
2. \( x + 2 = 0 \) ⟹ \( x = -2 \)

### Step 4: Determine the domain
The function \( f(x) \) is undefined at \( x = 3 \) and \( x = -2 \). Therefore, the domain of \( f(x) \) includes all real numbers except these two points. We can express the domain in interval notation as:
\[ \text{Domain of } f(x) = (-\infty, -2) \cup (-2, 3) \cup (3, \infty) \]

### Final Answer
The domain of the function \( f(x) \) is:
\[ \text{Domain} = \{ x \in \mathbb{R} \mid x \neq -2, x \neq 3 \} \]

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