The domain of `f(x) = (log_(2)(x+4))/(x^(2) + 3x + 2)` is

Options

A. R - {-1, -2}
B. `(-2, +oo)`
C. R - {-1, 2, -3}
D. `(-4, +oo), - {-1, -2}`

Answer

To find the domain of the function \( f(x) = \frac{\log_2(x + 4)}{x^2 + 3x + 2} \), we need to consider two main conditions:

1. The denominator must not be equal to zero.
2. The argument of the logarithm must be positive.

### Step 1: Analyze the Denominator

First, we need to ensure that the denominator \( x^2 + 3x + 2 \) is not equal to zero.

To find the values where the denominator is zero, we can factor the quadratic expression:

\[ x^2 + 3x + 2 = (x + 1)(x + 2) \]

Setting this equal to zero gives us:

\[ (x + 1)(x + 2) = 0 \]

Thus, the solutions are:

\[ x + 1 = 0 \quad \Rightarrow \quad x = -1 \]
\[ x + 2 = 0 \quad \Rightarrow \quad x = -2 \]

So, the denominator is zero at \( x = -1 \) and \( x = -2 \). Therefore, we have:

\[ x \neq -1 \quad \text{and} \quad x \neq -2 \]

### Step 2: Analyze the Logarithm

Next, we need to ensure that the argument of the logarithm \( x + 4 \) is positive:

\[ x + 4 > 0 \]

This simplifies to:

\[ x > -4 \]

### Step 3: Combine the Conditions

Now, we need to combine the two conditions:

1. \( x > -4 \)
2. \( x \neq -1 \) and \( x \neq -2 \)

The first condition \( x > -4 \) defines an interval from \( -4 \) to \( +\infty \). The second condition excludes the points \( -1 \) and \( -2 \) from this interval.

### Step 4: Determine the Domain

The interval \( (-4, +\infty) \) excludes the points \( -1 \) and \( -2 \). Thus, we can express the domain of \( f(x) \) as:

\[ \text{Domain of } f(x) = (-4, -2) \cup (-2, -1) \cup (-1, +\infty) \]

### Final Answer

The domain of the function \( f(x) \) is:

\[ (-4, -2) \cup (-2, -1) \cup (-1, +\infty) \]

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