I should decide if these functions belong to Schwartz space $\Bbb S $:
$f_1(x) = \frac {1}{1+|x|^4}$
$f_2(x)= e ^{-|x-3|^4}$
$f_3(x)= e ^{-|x|}$
The Schwartz space is a space where the functions decrease rapidly at $\infty$ and it is a subspace of $C^{\infty}$
I think that $f_1$ might not be decreasing enough, but it is continuous and when I graphed it, it looked like the distribution.
$f_2$ probably is in $\Bbb S$ because the power of the function is even so the derivatives don't have any singularity.
$f_3$ is not in $\Bbb S$, there is a discontinuity for the first derivative at zero.
Are my thoughts about Schwartz space correct please?