$F : \mathbb{Z} \to \mathbb{Z}$, $F(n) = 2 -3n$. Is $F$ one-to-one? Onto?

Question

Define $F : \mathbb{Z} \to \mathbb{Z}$ by the rule $F(n) = 2 -3n$, for all $n \in \mathbb{Z}$. Is $F$ one-to-one? Onto?

Now, I understand that one-to-one means that nothing in the co-domain is being pointed to twice. I also understand onto; which means that every point in the codomain is being pointed to by a point in the domain. Beyond that, I am unsure of where to start here, in terms of proving or disproving them.

Answer 1

Hints:

• Suppose $F(m) = F(n)$, that is $2-3m = 2-3n$; what can you say about the relationship between $m$ and $n$?
• Let $k \in \mathbb{Z}$. Can you solve $f(n) = k$? First solve the equation $2 - 3n = k$ for $n$, then check to see whether or not $n \in \mathbb{Z}$.