Find the domain of the following functions.
`f(x)=log_(3+x)(x^(2)-1)`

Answer

To find the domain of the function \( f(x) = \log_{3+x}(x^2 - 1) \), we need to ensure that both the base of the logarithm and the argument of the logarithm satisfy certain conditions.

### Step 1: Identify the conditions for the logarithm

The logarithm \( \log_b(a) \) is defined under two conditions:
1. The base \( b \) must be greater than 0 and not equal to 1.
2. The argument \( a \) must be greater than 0.

In our case:
- The base \( b = 3 + x \)
- The argument \( a = x^2 - 1 \)

### Step 2: Set up inequalities based on the conditions

1. For the base \( 3 + x \):
- \( 3 + x > 0 \)
- \( 3 + x \neq 1 \)

2. For the argument \( x^2 - 1 \):
- \( x^2 - 1 > 0 \)

### Step 3: Solve the inequalities

**Inequality 1: \( 3 + x > 0 \)**

\[ x > -3 \]

**Inequality 2: \( 3 + x \neq 1 \)**

\[ x \neq -2 \]

**Inequality 3: \( x^2 - 1 > 0 \)**

This can be factored as:

\[ (x - 1)(x + 1) > 0 \]

To solve this inequality, we find the critical points, which are \( x = -1 \) and \( x = 1 \). We can test the intervals determined by these points:

- For \( x < -1 \) (e.g., \( x = -2 \)): \( (-2 - 1)(-2 + 1) = (-3)(-1) > 0 \) (True)
- For \( -1 < x < 1 \) (e.g., \( x = 0 \)): \( (0 - 1)(0 + 1) = (-1)(1) < 0 \) (False)
- For \( x > 1 \) (e.g., \( x = 2 \)): \( (2 - 1)(2 + 1) = (1)(3) > 0 \) (True)

Thus, the solution to \( x^2 - 1 > 0 \) is:

\[ x < -1 \quad \text{or} \quad x > 1 \]

### Step 4: Combine the conditions

Now we combine the results from the inequalities:

1. From \( x > -3 \)
2. From \( x \neq -2 \)
3. From \( x < -1 \) or \( x > 1 \)

The intervals we have are:
- From \( x > -3 \): This gives us \( (-3, \infty) \)
- From \( x < -1 \): This gives us \( (-\infty, -1) \)
- From \( x > 1 \): This gives us \( (1, \infty) \)

### Step 5: Exclude the point \( x = -2 \)

Now, we need to exclude \( x = -2 \) from our intervals. Thus, the domain of the function is:

\[ (-3, -2) \cup (-2, -1) \cup (1, \infty) \]

### Final Answer

The domain of the function \( f(x) = \log_{3+x}(x^2 - 1) \) is:

\[ \boxed{(-3, -2) \cup (-2, -1) \cup (1, \infty)} \]