finding a bijection with conditions

Question

$x$ and $y$ are variables in the set $\mathbb R$. With bijective functions $f$ and $g$, $x$ and $y$ respectively goes to $a$ and $b$ which is in the set $[-1, 1]$. So, $f(x)=a$ and $g(y)=b$.For example, $f$ and $g$ can be hyperbolic tangent function.

Is there $f$ and $g$ that satisfies the followings?

1. The set $xy=1 ~(x, y>0)$ is equal to the set $a+b=1$
2. The set $x+y=0$ is equal to the set $a+b=0$

Answer 1

$$x + y = 0\implies f(x) + g(-x) = 0\implies g(x) = -f(-x).$$ $$xy = 1\implies f(x) + g(1/x) = 1\implies f(x) - f(-1/x) = 1.$$ This suggests $$f(x) = - x/(x + 1)\hbox{ for } x \ge 0;$$ $$f(x) = x/(x - 1)\hbox{ for } x < 0;$$ $$g(x) = -f(-x).$$