Bijective mapping from the integers to the rationals [duplicate]

Question

Definition

A rational is a number that can be expressed as A/B where A and B are integers, B is positive, and A and B are co-prime.

Task

You are to write two programs/functions, one which takes a rational and output an integer, another which takes an integer and outputs a rational.

Denote the two functions as f and g respectively.

They are meant to be inverses of each other, meaning that f(g(n)) = n and g(f(r)) = r for all integers n and all rationals r.

f(r) must be defined for all rationals r and g(n) for all integers n.

f(a) == f(b) if and only if a == b; g(p) == g(q) if and only if p == q.

Specs

• You can take the rational and the integer in any sensible format.
• You can choose different format for f and g, but the format must be consistent within f and within g.
• You can assume that the rational input is simplified.
• You do not need to simplify the rational output.
• The two functions/programs must be stand-alone.

Testcases

Below are valid rational inputs:

0/1
1/1
5/7
-1/3

Below are invalid rational inputs:

2/4 (can be output)
0/0
3/0
3/-4
-1/-5

Note

1/2 and 2/4 must map to the same integer, although only the former will be supplied.

As always, if my problem is not clear enough, please address in the comments.