Question

Let f(x) and g(x) be two periodic functions defined on R. The period of f(x) is 4, the period of g(x) is 2, and f(x) is odd function. When x $\in$ (0,2), $\rm f(x) = \sqrt{1 - (x - 1) {}^{2} }$ , $\rm g(x) =\begin{cases} \rm k(x + 2) , 0 < x \leq1 \\ \rm - \frac{1}2 \: \: \: \: \: \: \: \: \: \: \: , 1 < x \leq2 \end{cases} $ , where k>0. if the equation f(x)=g(x) on the interval (0,9] has 8 different real roots, then the value range of k is ______.​