1. The problem asks us to determine which statement supports the claim that the function $f(x) = x^3 + x$ is an odd function and not an even function. 2. Recall the 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. We will evaluate $f(-3)$ and compare it to $f(3)$ and $-f(3)$: $$f(3) = 3^3 + 3 = 27 + 3 = 30$$ $$f(-3) = (-3)^3 + (-3) = -27 - 3 = -30$$ 4. Check if $f(-3) = f(3)$: $$-30 \neq 30$$ So, $f$ is not even. 5. Check if $f(-3) = -f(3)$: $$-30 = -30$$ This is true, so $f$ is odd. 6. Now check the options: - A: $f(0) = -f(0)$ means $f(0) = 0$, which is true but does not prove oddness. - B: $-f(3) = f(3)$ means $-30 = 30$, false. - C: $f(-3) = f(3)$ means $-30 = 30$, false. - D: $f(-3) = -f(3)$ means $-30 = -30$, true and supports oddness. **Final answer:** Option D is true and supports that $f$ is an odd function.