How to find the range of the following function?

Question

I have following function $$f(x_1,x_2)=\frac{(x_1,x_2)}{x_1+x_2+1}$$ where domain of $f$ is $\{(x_1,x_2)|x_1+x_2+1>0\}$. How to find the range of this function?

Answer 1

Let $D$ be the domain of $f.$ For $b>-1,$ let $L_b$ be the line $x+y=b.$ Then $D=\bigcup_{b>-1}L_b.$

Now if $(x,y) \in L_b, b>-1,$ then

$$f(x,y) =\frac{(x,y)}{b+1}.$$

Because $x/(b+1)+y/(b+1) = (x+y)/(b+1) = b/(b+1),$ we see that $f(L_b)\subset L_{b/(b+1)}.$ In fact $f$ is a bijection between $L_b$ and $L_{b/(b+1)}.$ Thus

$$\tag 1 f(D) = \bigcup_{b>-1}f(L_b) = \bigcup_{b>-1}L_{b/(b+1)}.$$

Observe $b\to b/(b+1)$ maps $(-1,\infty)$ bijectively onto $(-\infty,1).$ Thus the union on the right of $(1)$ is the set $x+y<1.$