1. **Problem statement:** Determine whether the function is even, odd, or neither, and find the zeroes of the function for part (a): $f(x) = 2x^3 - 4x$. 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 $x$. - Otherwise, the function is neither even nor odd. 3. **Check evenness or oddness:** Calculate $f(-x)$: $$f(-x) = 2(-x)^3 - 4(-x) = 2(-x^3) + 4x = -2x^3 + 4x$$ Compare with $f(x)$ and $-f(x)$: - $f(x) = 2x^3 - 4x$ - $-f(x) = -2x^3 + 4x$ Since $f(-x) = -f(x)$, the function is **odd**. 4. **Find zeroes:** Set $f(x) = 0$: $$2x^3 - 4x = 0$$ Factor out $2x$: $$2x(x^2 - 2) = 0$$ Set each factor to zero: - $2x = 0 \Rightarrow x = 0$ - $x^2 - 2 = 0 \Rightarrow x^2 = 2 \Rightarrow x = \pm \sqrt{2}$ 5. **Final answer:** - The function $f(x) = 2x^3 - 4x$ is **odd**. - The zeroes are $x = 0$, $x = \sqrt{2}$, and $x = -\sqrt{2}$.