Find the domain of the following functions.
`f(x)=(log_(2)(x+3))/((x^(2)+3x+2))`

Answer

To find the domain of the function \( f(x) = \frac{\log_2(x + 3)}{x^2 + 3x + 2} \), we need to ensure that both the numerator and the denominator are defined.

### Step 1: Analyze the Denominator

The denominator is \( x^2 + 3x + 2 \). For the function to be defined, the denominator must not be equal to zero.

1. Set the denominator equal to zero:
\[ x^2 + 3x + 2 = 0 \]

2. Factor the quadratic:
\[ (x + 1)(x + 2) = 0 \]

3. Solve for \( x \):
\[ x + 1 = 0 \quad \Rightarrow \quad x = -1 \]
\[ x + 2 = 0 \quad \Rightarrow \quad x = -2 \]

Thus, the values \( x = -1 \) and \( x = -2 \) must be excluded from the domain.

### Step 2: Analyze the Numerator

The numerator is \( \log_2(x + 3) \). For the logarithm to be defined, its argument must be greater than zero.

1. Set the argument of the logarithm greater than zero:
\[ x + 3 > 0 \]

2. Solve for \( x \):
\[ x > -3 \]

### Step 3: Combine the Conditions

Now, we combine the conditions obtained from the numerator and denominator:

1. From the numerator, we have \( x > -3 \).
2. From the denominator, we must exclude \( x = -1 \) and \( x = -2 \).

### Step 4: Write the Domain

The domain of the function \( f(x) \) can be expressed in interval notation. Since \( x \) must be greater than \(-3\) but cannot be \(-1\) or \(-2\), we can write the domain as:

\[ \text{Domain of } f(x) = (-3, -2) \cup (-2, -1) \cup (-1, \infty) \]

### Final Answer

Thus, the domain of the function \( f(x) = \frac{\log_2(x + 3)}{x^2 + 3x + 2} \) is:
\[ (-3, -2) \cup (-2, -1) \cup (-1, \infty) \]

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