If $ | \int \phi(x) f(x) dx| \le 2 \sup_x (1+\|x \|^2)^M |\phi(x)|$, then $f$ is a tempered distribution

I am looking to see if the following condition is true: Suppose that$$ \left| \int \phi(x) f(x) dx \right| \le C \sup_x (1+\|x \|^2)^M |\phi(x)|$$for all Schwartz-class functions $\phi$ and some integer $M>0$ and constant $C$, then $f$ is tempered distribution.

I would also appreciate a reference.