# 6. Solution:

Let's find the range of the function $z=g(x)=4.5-2 \cos x+\cos 2 x$. The function $g(x)$ is defined for all real numbers. We will make a substitution. Let $t=\cos x$. Then $z=4.5-2 t+(2 t^{2}-1)=3.5+2(t^{2}-t)=3+2(t-0.5)^{2}$ for $t \in[-1 ; 1]$, and $E_{g}=[3 ; 7.5]$. The function $u=\frac{\pi}{9} z$ takes all values from the interval $\left[\frac{\pi}{3} ; \frac{5 \pi}{6}\right]$. The range of the function $f(x)$ coincides with the range of the function $y=\sin u$, where $u \in\left[\frac{\pi}{3} ; \frac{5 \pi}{6}\right]$. Therefore, $E_{f}=[0.5 ; 1]$
Answer: $E_{f}=[0,5 ; 1]$.

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