Given a function $f: \mathbf{N}_0 \to \mathbf{N}_0$, defined $$ f(x) = \begin{cases} x+3 & \text{if } x \in \mathbf{N}_{\text{even}} \\ x-1 & \text{if } x \in \mathbf{N}_{\text{odd}} \end{cases} $$
How can I find the preimage $f^{-1} ({1,2,3,4})$ ?
Any help would be appreciated
From the definition $$f(x) = \begin{cases} x+3 & \text{if } x \in \mathbf{N}_{\text{even}} \\ x-1 & \text{if } x \in \mathbf{N}_{\text{odd}} \end{cases}$$ we get that odd numbers map to even numbers and even numbers map to odd numbers.
Since $f(x)=1$ is odd, $x$ is even. Thus $$f(x)=x+3=1$$ Thus the pre-image of $1$ does not exist in $N$
Similarly we find $$f^{-1}(2)=3, f^{-1}(3)=0 , f^{-1}(4)=5.$$ Therefore,$$ f^{-1} ({1,2,3,4})=(?,3,0,5)$$