You need to prove that is (a) an injection (one-to-one); and (b) a surjection (onto function).
For (a), you need to show that if and , then . You'll need to consider the cases where and take various combinations of positive and negative signs.
To start you off, suppose , and . Then ...?
Then look at what happens if and are both negative. Finally show that if and have different signs, then .
For (b) you need to show that for any , we can find an such that . Again you'll need to consider two separate cases: this time it will be whether is even or odd.
Let us know if you can't see how to complete it from here.