Doman and range of a simple relation

Question

Relation xRy if x≥y^2 (on real numbers),

I'm assuming the domain is (o, infinity) and the range is all real numbers?

Answer 1

If we set $R=\{ (x, y) \in \mathbb{R} | x \geq y^2 \}$ then the domain is $D= \{ x | (x,y) \in R \} $ and the range is $I= \{ y | (x,y) \in R \} $.

Then $D=[0, + \infty)$. In fact $D \supseteq \{ x^2 | (x^2,x) \in R , x \in \mathbb{R} \} = [0, + \infty)$, and $D \subseteq [0, + \infty)$ since $\forall x \in D$ there exists $y \in \mathbb{R}$ such that $x \geq y^2$, so $x \in [0, + \infty)$.

In a similar way one can prove that $I = \mathbb{R}$.