smc

From the answer choices, note that $$\begin{array}{rl}f(25)& =f\left(\frac{25}{11}\cdot 11\right)\\ & =f\left(\frac{25}{11}\right)+f(11)\\ & =f\left(\frac{25}{11}\right)+11.\end{array}$$ On the other hand, we have $$\begin{array}{rl}f(25)& =f(5\cdot 5)\\ & =f(5)+f(5)\\ & =5+5\\ & =10.\end{array}$$ Equating the expressions for $f(25)$ produces $$f\left(\frac{25}{11}\right)+11=10,$$ from which $f\left(\frac{25}{11}\right)=-1.$ Therefore, the answer is $\boxed{\frac{25}{11}}.$ Remark Similarly, we can find the outputs of $f$ at the inputs of the other answer choices: \begin{alignat}{10} &\textbf{(A)} \qquad && f\left(\frac{17}{32}\right) \quad && = \quad && 7 \\ &\textbf{(B)} \qquad && f\left(\frac{11}{16}\right) \quad && = \quad && 3 \\ &\textbf{(C)} \qquad && f\left(\frac{7}{9}\right) \quad && = \quad && 1 \\ &\textbf{(D)} \qquad && f\left(\frac{7}{6}\right) \quad && = \quad && 2 \end{alignat} Alternatively, refer to Solutions 2 and 4 for the full processes.