Hi everyone.
I've got a one task to do i need to show if the function g: R -> R^2 by g(x) = (x+1, x+2) is g onto?
I think is not but i do not really know how to show this understanding.
I'll be appreciate for any help.
Regards
ok if you see the diagram you may better understand. the domain is a line. the codomain is a plain and the range is a straight line in that plane. now is that range line the same as full plane ?? how can that be ... this is a good way of illuminating things. keep track of pictures.
Simply assume that there exists $\displaystyle x \in {R}$ such that $\displaystyle g(x) = (0,0)$. This gives us:
(I) $\displaystyle x+1 = 0$
(II) $\displaystyle x+2 = 0$
$\displaystyle \Rightarrow x+1 = x+2 \Rightarrow 1 = 2 \Rightarrow$ contradiction.
So there is no $\displaystyle x \in {R}$ such that $\displaystyle g(x) = (0,0)$ and thus g is not onto