 Solution

In the final state, the second-register amplitude vector associated with outcome $y$ isFor injective $f$, part (ii) says that its squared norm is $2^{-2n}{2}^n=2^{-n}$. By the Born rule for a product-basis measurement, every $y\in B_n$ is therefore observed with probability $2^{-n}$: the input register is uniformly distributed.