9. (16 points) If the function
$$
f(x)=256 x^{9}-576 x^{7}+432 x^{5}-120 x^{3}+9 x \text {, }
$$

find the range of the function $f(x)$ for $x \in[-1,1]$.
II. 9. Note that,
$$
\begin{array}{l}
\cos 3 \theta=\cos 2 \theta \cdot \cos \theta-\sin 2 \theta \cdot \sin \theta \\
=\left(2 \cos ^{2} \theta-1\right) \cos \theta-2\left(1-\cos ^{2} \theta\right) \cos \theta \\
=4 \cos ^{3} \theta-3 \cos \theta .
\end{array}
$$

Therefore, $\cos 9 \theta=4 \cos ^{3} 3 \theta-3 \cos 3 \theta$
$$
\begin{aligned}
= & 4\left(4 \cos ^{3} \theta-3 \cos \theta\right)^{3}-3\left(4 \cos ^{3} \theta-3 \cos \theta\right) \\
= & 256 \cos ^{9} \theta-576 \cos ^{7} \theta+432 \cos ^{5} \theta- \\
& 120 \cos ^{3} \theta+9 \cos \theta .
\end{aligned}
$$

Let $x=\cos \theta \in[-1,1](\theta \in \mathbf{R})$. Then
$$
f(x)=\cos 9 \theta \in[-1,1] \text {. }
$$

Therefore, the range of the function $f(x)$ is $[-1,1]$.