Inverse Fourier of $f(\frac{-x}{a}+b)$

Question

I need to calculate Inverse Fourier of $\exp({-\frac{w^2}{a}+wi})$. I used the formula of Fourier Transform of $f(\frac{-x}{a}+b)$ as $$a\exp({-iaby})g(-ay)$$ From $g(-ay)$, I could find $f(-ax)$, however I am not sure how to connect this to $f(\frac{-x}{a}+b)$. Thanks