6. Find the set of values of the function $f(x)=4 \cos \left(\frac{\pi}{3} \sin \left(x^{2}+6 x+10-\sin x\right)\right)$.

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# Solution:

$f(x)=4 \cos \left(\frac{\pi}{3} \sin \left(x^{2}+6 x+10-\sin x\right)\right)=4 \cos \left(\frac{\pi}{3} \sin \left((x+3)^{2}+1-\sin x\right)\right)$. Since $g(x)=(x+3)^{2}+1-\sin x \in[0 ; \infty)$, and the range of this function contains a numerical ray, then $\varphi(x)=\sin \left((x+3)^{2}+1-\sin x\right) \in[-1 ; 1], \quad E_{\varphi}=[-1 ; 1], \psi(x)=\frac{\pi}{3} \sin \varphi(x) \in[-\pi / 3 ; \pi / 3]$, $E_{\psi}=[-\pi / 3 ; \pi / 3]$. Therefore, $\cos \left(\frac{\pi}{3} \sin \left((x+3)^{2}+1-\sin x\right)\right) \in[1 / 2 ; 1] \Rightarrow 4 \cos \left(\frac{\pi}{3} \sin \left((x+3)^{2}+1-\sin x\right)\right) \in[2 ; 4], E_{f}=[2 ; 4]$.

Answer: $[2 ; 4]$.