I am currently working on a task and I have a problem doing the next step: Given a function $f:\{0,1\}^n\to\{0,1\}^n$, such that for each distinct $x\in\{0,1\}^n$ the output $f(x)$ is unique, except for $f(0^n)=f(1^n)$, I am trying to compute the inner product $\left\langle\psi_y\middle|\:\psi_y\right\rangle$ for a fixed $y\in\{0,1\}$ and $|\psi_y\rangle=\frac{1}{2^n}\sum\limits_{x\in\{0,1\}^n}(-1)^{x\cdot y}|f(x)\rangle$.
I started computing the inner product by expanding it using the tensor-product. Then I get the equation
$\left\langle\psi_y|\psi_y\right\rangle=\frac{1}{2^{2n}}\left\langle\sum\limits_{x_1\in\{0,1\}}(-1)^{x_1\cdot y_1}\langle f(x_1)|\middle|\:\sum\limits_{x_1\in\{0,1\}}(-1)^{x_1\cdot y_1}|f(x_1)\rangle\right\rangle\otimes\cdots\otimes\left\langle\sum\limits_{x_n\in\{0,1\}}(-1)^{x_n\cdot y_n}\langle f(x_n)|\middle|\:\sum\limits_{x_n\in\{0,1\}}(-1)^{x_n\cdot y_n}|f(x_n)\rangle\right\rangle\\=\frac{1}{2^{2n}}\left\langle\sum\limits_{x_1\in\{0,1\}}(-1)^{x_1\cdot y_1}\langle f(x_1)|\middle|\:\sum\limits_{x_1\in\{0,1\}}(-1)^{x_1\cdot y_1}|f(x_1)\rangle\right\rangle\cdot...\cdot \left\langle\sum\limits_{x_1\in\{0,1\}}(-1)^{x_1\cdot y_1}\langle f(x_1)|\middle|\:\sum\limits_{x_1\in\{0,1\}}(-1)^{x_1\cdot y_1}|f(x_1)\rangle\right\rangle$
However, at this point I am not really sure how to move on. The result is supposed to be $\frac{1}{2^n}+\frac{1}{2^{2n-1}}$ or $\frac{1}{2^n}-\frac{1}{2^{2n-1}}$ depending on $y$.
I hope someone can give me a hint or tell me what the next step could be. Maybe I already did something wrong.