Question: determine whether the function is an even or odd function or neither: $f(x)=4x^3$
1. **State the problem:** Determine if the function $f(x)=4x^3$ is even, odd, or neither.
2. **Recall definitions:**
- A function $f$ is **even** if $f(-x) = f(x)$ for all $x$.
- A function $f$ is **odd** if $f(-x) = -f(x)$ for all $x$.
3. **Calculate $f(-x)$:**
$$f(-x) = 4(-x)^3 = 4(-x^3) = -4x^3$$
4. **Compare $f(-x)$ with $f(x)$ and $-f(x)$:**
- $f(x) = 4x^3$
- $-f(x) = -4x^3$
Since $f(-x) = -4x^3 = -f(x)$, the function satisfies the condition for being odd.
5. **Conclusion:**
The function $f(x) = 4x^3$ is an **odd function** because $f(-x) = -f(x)$ for all $x$.
**Final answer:** $f(x) = 4x^3$ is an odd function.