Interprete $f(L_2)$ and $f(L_3)$.

Question

1. Let $f(z)=z^2$ for $z \in \mathbb{C} $, $L_2$ = {$(0,y)$:y $\geqq0$}. Interprete $f(L_2)$

2. Let $f(z)=iz$ for $z \in \mathbb{C}$, $L_3$ = {$(x,y)$:$y=x+1$. Interprete $f(L_3)$.

I don't know how to start, please help me how to answer this one. I'm stuck.

Answer 1

I show you 2.:

Let $w \in f(L_3)$. Then $w=iz$ for some $z \in L_3$. Hence $z=x+i(x+1)$ and therefore

$w=iz=i(x+i(x+1))=-(x+1)+ix$.

This gives $f(L_3) \subseteq L:=\{(-(x+1),x): x \in \mathbb R\}$.

$L$ is a line in $ \mathbb R^2$ !

It is your turn to show, that we also have $f(L_3) \supseteq L$.

Consequence: $f(L_3) = L$.