Problem 97

Since the function $f(x) = \begin{cases} 2^{1-x}, & x \leqslant 1 \ \log_{4}(x-1), & x > 1 \end{cases}$,

We have $f(9)=\log_{4}8= \frac {3}{2}$,

And $f(\log_{2} \frac {1}{6})=2^{1-\log_{2} \frac {1}{6}}=2\div \frac {1}{6}=12$,

Therefore, $2f(9)+f(\log_{2} \frac {1}{6})=2\times \frac {3}{2}+12=\boxed{15}$.

To find the answer, we first calculate $f(9)=\log_{4}8= \frac {3}{2}$ and $f(\log_{2} \frac {1}{6})=2^{1-\log_{2} \frac {1}{6}}=12$. Then, we add these results together to obtain the final answer.

This problem focuses on finding the values of a given function. It is important to carefully read the problem and make proper use of the function's properties when solving.