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

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# Solution:

$f(x)=2 \cos \left(\frac{\pi}{4} \sin \left(x^{2}+2 x+2+\cos x\right)\right)=2 \cos \left(\frac{\pi}{4} \sin \left((x+1)^{2}+1+\cos x\right)\right)$. Since $g(x)=(x+1)^{2}+1+\cos x \subset[0 ; \infty)$, and the range of this function contains a numerical ray, then

$\varphi(x)=\sin \left((x+1)^{2}+1+\cos x\right) \in[-1 ; 1], E_{\varphi}=[-1 ; 1], \psi(x)=\frac{\pi}{4} \sin \left((x+1)^{2}+1+\cos x\right) \in[-\pi / 4 ; \pi / 4]$,

$E_{\psi}=[-\pi / 4 ; \pi / 4]$. Therefore,

$\cos \left(\frac{\pi}{4} \sin \left((x+1)^{2}+1+\cos x\right)\right) \in[\sqrt{2} / 2 ; 1] \Rightarrow 2 \cos \left(\frac{\pi}{4} \sin \left((x+1)^{2}+1+\cos x\right)\right) \in[\sqrt{2} ; 2]$, $E_{f}=[\sqrt{2} ; 2]$

Answer: $[\sqrt{2} ; 2]$