Is there a mathematical function which flips 1 and 2? I'm looking for a function $f : \mathbb{Z} \rightarrow \mathbb{Z}$ such that $f(f(x)) = x$ and $f(1) = 2$. In particular, I don't want this function to be a piecewise function. Does such a function even exist?

For example, I can define the above function as follows:
\begin{equation}
f(x) = \begin{cases}
2 & x = 1 \\
1 & x = 2
\end{cases}
\end{equation}
However, I was wondering whether there's a way to define this function without writing conditions.

How about $f(x) = x-(-1)^x$? That works. 
