Graphically speaking what is the difference between:
$f:\mathbb{R}\rightarrow\mathbb{R}$ such that $f(x)=x^2$, and
$g:\mathbb{R}\rightarrow\mathbb{R}^2$ such that $g(x)=(x,x^2)$?
I know that $f$ just looks like a parabola $P=\{(x,y):y=x^2\}$ in $\mathbb{R}^2$ but doesn't the second also look like that since the function $g$ would just be returning to each point point $x$ on the $x$-axis, the point $(x,x^2)$ in $\mathbb{R}^2$. But it seems obviously wrong that they would be the same. I have read that for any function $f:\mathbb{R^n}\rightarrow\mathbb{R}$, the graph of $f$ is in $\mathbb{R}^{n+1}$ so I assume for $g$ the graph should actually be in 3-dimensions but why would we need the $z$-axis?
Similarly wouldn't the graph $h:P\subseteq\mathbb{R}^2 \rightarrow \mathbb{R}$ such that $h(x,x^2)=x$ just be the same parabola as $f$ but to each point $(x,x^2)$ of the parabola it returns the value on the $x$-axis. Why would this need to be 3-dimensional. I think I'm getting myself mixed up between functions and curves and can't stop confusing myself! I would really appreciate some help.