The domain of the function `f(x)=1/(9-x^2)+log_(20)(x^3-3x)` is

Options

A. `(-sqrt3,0)cup(sqrt(3),oo)`
B. `(-sqrt3,0)cup(sqrt(3),3)`
C. `(-sqrt3,0)cup(3,oo)`
D. `(-sqrt3,0)cup(sqrt(3),3)cup(3,oo)`

Answer

To find the domain of the function \( f(x) = \frac{1}{9 - x^2} + \log_{20}(x^3 - 3x) \), we need to consider the restrictions imposed by both terms in the function.

### Step 1: Analyze the first term \( \frac{1}{9 - x^2} \)

For this term to be defined, the denominator cannot be zero:

\[ 9 - x^2 \neq 0 \]

This implies:

\[ x^2 \neq 9 \implies x \neq 3 \quad \text{and} \quad x \neq -3 \]

### Step 2: Analyze the second term \( \log_{20}(x^3 - 3x) \)

For the logarithmic function to be defined, the argument must be greater than zero:

\[ x^3 - 3x > 0 \]

Factoring the expression:

\[ x(x^2 - 3) > 0 \]

This can be further factored as:

\[ x(x - \sqrt{3})(x + \sqrt{3}) > 0 \]

### Step 3: Determine the critical points

The critical points from the inequality \( x(x - \sqrt{3})(x + \sqrt{3}) = 0 \) are:

- \( x = 0 \)
- \( x = \sqrt{3} \)
- \( x = -\sqrt{3} \)

### Step 4: Test intervals

We will test the sign of \( x(x - \sqrt{3})(x + \sqrt{3}) \) in the intervals determined by these critical points:

1. **Interval \( (-\infty, -\sqrt{3}) \)**: Choose \( x = -2 \)
- \( (-2)(-2 - \sqrt{3})(-2 + \sqrt{3}) > 0 \) (positive)

2. **Interval \( (-\sqrt{3}, 0) \)**: Choose \( x = -1 \)
- \( (-1)(-1 - \sqrt{3})(-1 + \sqrt{3}) < 0 \) (negative)

3. **Interval \( (0, \sqrt{3}) \)**: Choose \( x = 1 \)
- \( (1)(1 - \sqrt{3})(1 + \sqrt{3}) < 0 \) (negative)

4. **Interval \( (\sqrt{3}, \infty) \)**: Choose \( x = 2 \)
- \( (2)(2 - \sqrt{3})(2 + \sqrt{3}) > 0 \) (positive)

### Step 5: Combine the results

From the analysis, we find that \( x(x - \sqrt{3})(x + \sqrt{3}) > 0 \) in the intervals:

- \( (-\infty, -\sqrt{3}) \)
- \( (\sqrt{3}, \infty) \)

### Step 6: Exclude points from the domain

Now, we need to exclude \( x = 3 \) and \( x = -3 \) from the domain:

- The interval \( (-\infty, -\sqrt{3}) \) does not include \( -3 \).
- The interval \( (\sqrt{3}, \infty) \) does not include \( 3 \).

### Final Domain

Thus, the domain of the function \( f(x) \) is:

\[ (-\infty, -\sqrt{3}) \cup (\sqrt{3}, 3) \cup (3, \infty) \]