Find the range of $f:[0,2\pi) \rightarrow \Bbb R, f(x)=(1+3^{\lfloor\frac{x}{\pi}\rfloor}|\sin x|)^\frac{1}{2}$. I need to check my solution for this problem: 
Let $f:[0,2\pi) \rightarrow \Bbb R, f(x)=(1+3^{\lfloor\frac{x}{\pi}\rfloor}|\sin x|)^\frac{1}{2}$. 
$1)$ Find the range of $f$. 
$2)$ Let $f_1=f\circ f$. Is $f_1$ a bijection? 
For the first one I got that the range is $[1,2]$. And for the second one, I think  that $f_1$ is not a bijection, since $f$ is not a bijection, because $|\sin x|$ is not injective. Is this correct? 

The given function is 
$$f(x)=\begin{cases}
\sqrt{(1+\sin x)} & \text{ if } 0 \leq x < \pi \\
\sqrt{(1-3\sin x)} & \text{ if } \pi \leq x < 2\pi
\end{cases}
$$
Now observe that
$f(f0))=f(f(\pi))$, hence the composition function is not bijective.
