What does it mean to write $(y,z)=G(x)$

Question

I understand this is a map from $\mathbb{R} \mapsto \mathbb{R^2}$.

Is it the case that $y=y(x)$ and $z=z(x)$? i.e

Are $y$ and $z$ individually functions of $x$ as well as being so jointly ($(y,z)=G(x)$)

Answer 1

When $G$ is map from $\mathbb R$ to $\mathbb R^2$, we write $G\colon \mathbb R\to\mathbb R^2$. This means that $G$ takes any real number $x\in\mathbb R$ and depending on $x$ returns an element of $\mathbb R^2$, i.e. a pair of real numbers $(y,z)$. For example \begin{align*} G\colon \mathbb R &\longrightarrow \mathbb R^2 \\ x &\longmapsto (2x, x+1) \end{align*} is such a map and we have $G(1)=(2,2)$, $G(2)=(4,3)$, $G(\pi) = (2\pi, \pi+1)$ etc.

Answer 2

if $\pi_1:(s,t) \mapsto s$ and $\pi_2:(s,t) \mapsto t$ are the two co-ordinate projection maps from your $\mathbb{R^2}$-frame, then the statement is saying that:

$$ y = \pi_1 \circ G(x) \\ z = \pi_2 \circ G(x) \\ $$

(emended, thanks to an error spotted by Christoph)