ILL 1979 Problem 27

For all rational $x$ satisfying $0 \leq x < 1$ , the functions $f$ is defined by $$f(x)=\begin{cases}\frac{f(2x)}{4},&\text{for }0 \leq x < \frac 12, \frac 34+ \frac{f(2x - 1)}{4}, & \text{for } \frac 12 \leq x < 1.\end{cases}$$ Given that $x = 0.b_1b_2b_3 \cdots$ is the binary representation of $x$ , find, with proof, $f(x)$ .