For each ordered pair ( a, b) of integers define a mapping alpha (a, b) : Z . Z by alapha a, b( n) = an + b.
( a) For which pairs ( a, b) is alpha (a, b) onto?
( b) For which pairs ( a, b) is alpha (a, b) one- to- one?
Is it $\displaystyle \alpha(a,b): \mathbb{Z} \rightarrow \mathbb{Z}$ defined by $\displaystyle \alpha(a,b)(n) = an+b$ ?
For (a), consider $\displaystyle |a|>1$ ...can $\displaystyle \alpha(a,b)$ be onto? Then examine remaining cases for $\displaystyle a$.
In (b), again concentrate on the value of $\displaystyle a$ (parameter $\displaystyle b$ acts like a shift, it can't affect whether the function is one-to-one or not). There are not many cases in which the function is not one-to-one, can you see them?