Depicting the Transformation O-gate

The following figure depicts the transformation gate $O_i$:

Let’s say, $i=0$. In that case, we’ll apply the function $f_0$. Per definition, $f_0(x)=0$. When we insert this into the above equation, we can see the output of $O_i$ is equal to its input:

$$O_0(|x\rangle\otimes|y\rangle)=|x\rangle\otimes|y\oplus|f_0(x)\rangle=|x\rangle\otimes|y\oplus|0\rangle=|x\rangle\otimes|y\rangle$$

We can safely state that not changing a state is reversible.

When $i=1$, we apply the function $f_1$, which returns 0 for $x=0$ and 1 for $x=1$. Thus, $f_1(x)=x$.

$$O_1(|x\rangle\otimes|y\rangle)=|x\rangle\otimes|y\oplus f_1(x)\rangle=|x\rangle\otimes|y\oplus x\rangle$$ ...