If I am correct , a parametric function/ curve in $2D$ is a function that takes all the elements of a set $T$ ( $\subset R$) as inputs and that sends back each time as output an element of $R^2$, that is a couple or a point $P=(X_P, Y_P)$ with $X_P= f(t)$ and $Y_P= g(t)$.
For example, if one sets $f(t)=t^2$ and $g(t)=t^3+1$, the corresponding parametric function defined , say, on $T= [0,10]$ should be ( at least if I am right as to the definition of a parametric function):
$\{(X_P,Y_P) |\space 0\leq t\leq 10 \space\&\space X_P=t^2 \space\& \space Y_P=t^3+1\}$.
When I ask Desmos to draw the graph of the parametric function $(f(t), g(t)$) I get a line that looks ( somehow ) like the $\sqrt(x)$ function.
But, when I try to recover the y=f(x) expression of the function , I find ( with $t\geq 0$)
$y=x^\frac{3}{2}+1 $, the graph of which looks roughly like the $x^2$ function.
So, is my definiton of a parametric function not correct? What do I miss?
Here a screenshot of Herbert Gross' calculus lectures which occasionned my research :
