To find the domain of the function \( f(x) = \frac{x}{16 - x^2} + \log_2(x^3 - 2x) \), we need to consider the restrictions imposed by both the rational part and the logarithmic part of the function.
### Step-by-Step Solution
1. **Identify the Rational Part:**
The rational part of the function is \( \frac{x}{16 - x^2} \). The denominator cannot be zero:
\[
16 - x^2 \neq 0
\]
This gives us:
\[
x^2 \neq 16 \implies x \neq 4 \quad \text{and} \quad x \neq -4
\]
2. **Identify the Logarithmic Part:**
The logarithmic part of the function is \( \log_2(x^3 - 2x) \). The argument of the logarithm must be greater than zero:
\[
x^3 - 2x > 0
\]
Factor the expression:
\[
x(x^2 - 2) > 0
\]
This can be factored further:
\[
x(x - \sqrt{2})(x + \sqrt{2}) > 0
\]
3. **Determine the Critical Points:**
The critical points from the inequality \( x(x - \sqrt{2})(x + \sqrt{2}) = 0 \) are:
\[
x = 0, \quad x = \sqrt{2}, \quad x = -\sqrt{2}
\]
4. **Test Intervals:**
We will test the sign of the expression in the intervals defined by the critical points:
- Interval \( (-\infty, -\sqrt{2}) \)
- Interval \( (-\sqrt{2}, 0) \)
- Interval \( (0, \sqrt{2}) \)
- Interval \( (\sqrt{2}, \infty) \)
- **For \( x < -\sqrt{2} \):** Choose \( x = -3 \):
\[
(-3)(-3 - \sqrt{2})(-3 + \sqrt{2}) > 0 \quad \text{(positive)}
\]
- **For \( -\sqrt{2} < x < 0 \):** Choose \( x = -1 \):
\[
(-1)(-1 - \sqrt{2})(-1 + \sqrt{2}) < 0 \quad \text{(negative)}
\]
- **For \( 0 < x < \sqrt{2} \):** Choose \( x = 1 \):
\[
(1)(1 - \sqrt{2})(1 + \sqrt{2}) < 0 \quad \text{(negative)}
\]
- **For \( x > \sqrt{2} \):** Choose \( x = 2 \):
\[
(2)(2 - \sqrt{2})(2 + \sqrt{2}) > 0 \quad \text{(positive)}
\]
5. **Combine the Results:**
From the sign analysis, we find that \( x(x - \sqrt{2})(x + \sqrt{2}) > 0 \) holds in the intervals:
\[
(-\infty, -\sqrt{2}) \quad \text{and} \quad (\sqrt{2}, \infty)
\]
6. **Final Domain:**
We must exclude \( x = 4 \) and \( x = -4 \) from the domain. Therefore, the domain of \( f(x) \) is:
\[
(-\infty, -\sqrt{2}) \cup (-4, -\sqrt{2}) \cup (\sqrt{2}, 4) \cup (4, \infty)
\]
### Final Answer:
The domain of \( f(x) \) is:
\[
(-\infty, -4) \cup (-\sqrt{2}, 0) \cup (\sqrt{2}, 4) \cup (4, \infty)
\]
