As an example consider $x(t)=t+\sin(t)$ and $y(t)=\cos(t)$ and the graph plotted between $t=0$ and $t=2\pi$.
The parametric curve looks like 2 half hills.
But if I work out $x^2 +y^2$ I get $t^2 + 2 t \sin(t) + \sin^2(t)$. This is always greater than one and so represents a circle of varying radius dependant on $t$. For a start why is the parametric equations not in one-to-one correspondence with the implicit function but also if I plot any of the implicit functions (for any $t$) then I do not get the same graph as the parametric graph.
I'm very confused. Would anyone be able to explain this?
Functions can be of either implicit or parametric representation. An implicit function is basically obtained by eliminating parameter t between them as in the cited example
$$x(t)= \cos^{-1} y(t)+ \sqrt{1-y(t)^2}$$
leaving no trace with any one to one mapped correspondences.
The parametrization of an implicit relation is by no means unique.
For example circle $x^2+y^2=1 $ has no unique parametrization. We have $ (x=\cos t, y=\sin t), (x= sech\, t, y=tanh\, t) $
But inter-mappings can be among parametric representations belonging to same curve.
But in any case mapping is not by finding distance to origin where parameter t is still available.