Given the functions $f(x)= \delta (x-a)$ $g(x)= \frac{1}{a} \delta \left(x- \frac{1}{a}\right)$ for a real constant $a\gt0$, is there a relationship between $f$ and $g$?
I believe that $ g\left(\frac{1}{x}\right)=af(x) $, but I'm not $100\%$ sure. I have used the properties of $\delta (f(x))$ with $f(x)= \frac{1}{x}-\frac{1}{a}$, which has the only root $x=a$, but I do not know what else to do in order to prove it.