3 questions over the function $f(x_0,x_1,x_2)=\frac{1}{1+x_0+x_1+x_2}(x_0+x_1,x_1+x_2,x_2+x_0).$

Question

Let $D=\{(x_0,x_1,x_2)\in\mathbb{R}^3|\ 1+x_0+x_1+x_2 =0\}$ the plane in $\mathbb{R}^3$ passing for the points $(-1,0,0)$, $(0,-1,0)$, $(0,0,-1)$ and let $E=\mathbb{R}^3-D$ its complement. Note that $E$ is a open set and is the union of two semispaces (separated for the plane $D$). Let $$f:E\to\mathbb{R}^3\ ,\ (x_0,x_1,x_2)\mapsto f(x_0,x_1,x_2)=\frac{1}{1+x_0+x_1+x_2}(x_0+x_1,x_1+x_2,x_2+x_0).$$

1. Show that for all $\vec{x}\in E$, $Df(\vec{x})$ is of maximal rank. (i.e no have critic points).

2. Show that $f$ is injective.

3. Characterized the images $f(E)$ how union of two semispaces separated for some plane.

Thanks for your help.

Next my advances:

1. For $(x_0,x_1,x_2)\in E$, I have that $$Df(x_0,x_1,x_2)=\frac{1}{(1+x_0+x_1+x_2)^2}\left( \begin{array}{ccc} 1+x_2& 1+x_2 &-x_0-x_1 \\ -x_1-x_2 &1+x_0 &1+x_0 \\ 1+x_1& -x_0-x_2& 1+x_1 \\ \end{array} \right)$$ but how show that $Df(x_0,x_1,x_2)\neq 0$?

2. For show that $f$ is injective it suffices to show that the determinant of $Df(x_0,x_1,x_2)$ is distinct of zero?

3. I not have idea how to perform this characterization.