6. Find the set of values of the function $f(x)=g\left(\sqrt{25-g^{2}(x)}\right)$, where $g(x)=|| x|-2|-1$.
Solution: The function $g(x)=|| x|-2|-1 \quad$ is defined on the entire number line and takes all values from the interval $[-1 ;+\infty)$. The graph of the function $g(x)$ is shown in the figure. The function $\phi(t)=\sqrt{25-t^{2}}$ is defined for $t \in[-5 ; 5]$. When $\quad t=g(x)$, the function $\quad \phi(t)=\sqrt{25-t^{2}}$ takes its values for $t \in[-1 ; 5]$, and the set of values of this function is the interval [0;5]. To find the set of values of the function $f(x)$, it is sufficient to find the set of values of the function $g(x)$ on the interval [0;5]. On the specified interval, $g(x)$ takes all values from the set $[-1 ; 2]$. Answer: $E_{f}=[-1 ; 2]$.