Determing if a parametric curve is smooth

Question

I have to determine whether the following curves are smooth or not and I'm having trouble with the following two functions:

Consider $f(t) = (t^{2}-1,t^{2}+1)^{T}$
The solution states: $f'(t) = (2t,2t)^{T}=$ 0 iff $t=0$. It is not a smooth curve as it cannot be written as graph of a function $y=f(x)$ or $x=f(y)$ near $(-1,1)^{T}$

Why is this? Is it not possible to write $y=x+2$?

The second function I am having trouble with is $f(t) = (t^{3}-1,t^{3}+1)^{T}$ as the solution states that the curve is smooth and can be written as $y=x+2$. Why is it possible in this case when the previous function could not be written like that?

Answer 1

Try visualizing what the curves look like for values of $t$ slightly less than $0$ and slightly greater than $0$: the difference between the two is that for the first one, you approach $(-1,1)$ and then turn back, while for the second one, you simply pass through the point, continuing along the straight line.

In other words, if you could use the $y = x+2$ trick for the first one, you should have all points $(x,x+2)$ lying on the curve for $x$ in a neighborhood of $-1$, including values of $x$ that are slightly below $-1$; this is not the case though, since the $x$-coordinate is always at least $-1$.