I've a doubt concerning parametrization of $f(x) = x^2$. I'm studying curves can be parametrized in different ways, changing the speed with which they travel their path. I'm pretty sure that if I'll write, $$ f(t)=(t,t^2)$$ and $$ f(t)=(2t,4t^2)$$ they represents the same curve travelled with different speed.
I'm not sure about it if we write something like,
$$ f(t)=(t^2,t^4)$$
Does the fact that the components of the latter aren't anymore multiples of the previous ones change something?
I know the question can seem a bit ambiguous and wide, but any explanation or advice on the topic is welcomed with enthusiasm!
Your notation is a bit muddy but your question is a good one.
To clear the notation, your first parameterization is $$\left\{ \matrix{ x=t\\f = t^2} \right. $$ and your second is $$ \left\{ \matrix{ x=2t\\f = 2t^2} \right. $$ and these are not equivalent curves; but if you meant $$ \left\{ \matrix{ x=2t\\f = (2t)^2} \right. $$ then those would be equivalent only with different "speeds" along the two parameterizations.
Your third curve $$ \left\{ \matrix{ x=t^2\\f = (t^{2})^{2} } \right. $$ would be the same except for one important fact: The range of $x$ is no longer $(-\infty,\infty)$, it is instead $(0,\infty)$. So this is a parameterization of only the left half of the original curve.
That is, you have the right idea but you need to be careful that the range of $x$ is fully covered.