How to find inverse of a relation if the inverse isn't a function?

Question

I am trying to find the inverse of the following function $f:\mathbb{Z}^+\rightarrow \mathbb{Z}$ given by $f(a)=\frac{(-1)^a(2a-1)+1}{4}$.

I switched $x$ and $y$ and then tried solving for $y$. This is the technique that everyone learned back in middle school/high school. However, since the inverse relation isn't a function, it is not possible. How do I find the inverse using a different method?

Answer 1

Here, the inverse function exists :

$$g(x)=2n$$ for $n>0$ and $$g(x)=-2n+1$$ for $n\le 0$

$f$ is actually a function from $\mathbb Z^+$ to $\mathbb Z$

Answer 2

$f$ acts on $\mathbb{N}\setminus\{0\}$ as follows: $$ f(n) = \left\{\begin{array}{rcl}\frac{n}{2} &\text{if}& n\equiv 0\pmod{2}\\-\frac{n-1}{2}&\text{if}&n\equiv 1\pmod{2} \end{array}\right.$$ hence $f^{-1}$ maps positive integers into even numbers and non-positive integers into odd numbers: $$ f^{-1}(m) = \left\{\begin{array}{rcl}2m &\text{if}& m>0\\1-2m&\text{if}&m\leq 0. \end{array}\right.$$