How to show that a function defines a tempered distribution and compute the Fourier transform

Question

Consider the following functions : $f(x) = 1 + x \cos(3x)$ and $g(x) = x \sin^2(x)$. How can I show that $f$ and $g$ define tempered distributions on $\mathbb{R}$ and compute their Fourier transforms (in the sense of tempered distributions)? Unfortunately, I have no idea how to proceed. Any hint will be much appreciated.

Thanks for your help.