Range of $\frac{x-y}{1+x^2+y^2}=f(x,y)$ I have a function $\frac{x-y}{1+x^2+y^2}=f(x,y)$. And, I want to find the range of it. I analyzed this function by plotting it on a graph and found interesting things. Like if the level curve is $0=f(x,y)$, then I get $y=x$ which is a linear function. But if the level curve is something not 0, then the level curve becomes a circle. And for big values of level curves, the circle disappears. Is there something I can use to find the range of this function?

For an alternative approach...
Notice $f(1,1)=0$ so $0$ is in the range of $f$. Now if $C\neq 0$ then $$f(x,y)=C \iff\Big(x-\frac{1}{2C}\Big)^2+\Big(y+\frac{1}{2C}\Big)^2=\frac{1}{2C^2}-1$$ The above relation contains at least one point in the $(x,y)-$plane iff $C\in \big[-\frac{1}{\sqrt{2}},0)\cup (0,\frac{1}{\sqrt{2}}\big]$. Putting everything together we see the range is $\Big[-\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}}\Big]$.
