How to determine this and please provide explanation if possible.
sure. a. not $\displaystyle 1-1$, ie $\displaystyle 1 \mapsto 1$ and $\displaystyle 2 \mapsto 1$ b/c $\displaystyle 1^3=1 \in \mathbb{Z}_7$ and $\displaystyle 2^3=1 \in \mathbb{Z}_7$
not onto. check $\displaystyle 0,1,2,3,4,5,6$
b. yes, it is $\displaystyle 1-1$, easily can be checked, check $\displaystyle 0,1,2,3,4,5,6,7,8,9,10$.
yes, it is onto. check $\displaystyle 0,1,2,3,4,5,6,7,8,9,10$.