To find the domain of the function \( f(x) = \frac{1}{9 - x^2} + \log_{20}(x^3 - 3x) \), we need to consider the conditions under which each part of the function is defined.
### Step 1: Analyze the first term \( \frac{1}{9 - x^2} \)
The expression \( \frac{1}{9 - x^2} \) is defined as long as the denominator is not zero. Therefore, we need:
\[
9 - x^2 \neq 0
\]
This leads to:
\[
x^2 \neq 9
\]
Taking the square root of both sides gives:
\[
x \neq 3 \quad \text{and} \quad x \neq -3
\]
### Step 2: Analyze the second term \( \log_{20}(x^3 - 3x) \)
The logarithmic function is defined only for positive arguments. Thus, we require:
\[
x^3 - 3x > 0
\]
Factoring the expression gives:
\[
x(x^2 - 3) > 0
\]
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:
\[
x = 0, \quad x = \sqrt{3}, \quad x = -\sqrt{3}
\]
### Step 4: Test intervals
We will test the sign of the expression \( x(x - \sqrt{3})(x + \sqrt{3}) \) in the intervals defined by these critical points:
1. **Interval \( (-\infty, -\sqrt{3}) \)**: Choose \( x = -2 \):
\[
(-2)(-2 - \sqrt{3})(-2 + \sqrt{3}) > 0 \quad \text{(positive)}
\]
2. **Interval \( (-\sqrt{3}, 0) \)**: Choose \( x = -1 \):
\[
(-1)(-1 - \sqrt{3})(-1 + \sqrt{3}) < 0 \quad \text{(negative)}
\]
3. **Interval \( (0, \sqrt{3}) \)**: Choose \( x = 1 \):
\[
(1)(1 - \sqrt{3})(1 + \sqrt{3}) < 0 \quad \text{(negative)}
\]
4. **Interval \( (\sqrt{3}, \infty) \)**: Choose \( x = 2 \):
\[
(2)(2 - \sqrt{3})(2 + \sqrt{3}) > 0 \quad \text{(positive)}
\]
### Step 5: Combine 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 the points from the first term
Now, we need to combine these intervals with the restrictions from the first term:
- Exclude \( x = 3 \) and \( x = -3 \) from the domain.
The valid intervals for the domain of \( f(x) \) are:
1. From \( (-\infty, -\sqrt{3}) \) (excluding -3)
2. From \( (\sqrt{3}, 3) \) (excluding 3)
3. From \( (3, \infty) \)
### Final Domain
Thus, the domain of the function \( f(x) \) is:
\[
(-\infty, -\sqrt{3}) \cup (-3, 0) \cup (\sqrt{3}, 3) \cup (3, \infty)
\]
