For this to be injective you want to make sure two different values from Z go to different values of N [i.e. if x ≠ y then f(x)≠f(y)]. That is equivalent to saying if f(x)=f(y) then x=y (the contrapositive).
Say (|4x+1| +1)/2 = (|4y+1| +1)/2 then you must prove x=y.
But this reduces down to |4x+1| = |4y+1|.
Now do this in cases:
1. If the insides of both absolute values are positive you get 4x+1=4y+1
2. If both negative then (what?)
3. If left inside is the only negative then -4x-1= 4y+1
but this gives -x -y = 1/2 (this can't be because (what?))
4. If right inside is negative then (what?)
I left out a few steps. Can you finish it?
Let me know by commenting back.