How to obtain the time reversal property in terms of f in hertz (from the formula in terms of $ \omega $)
To obtain X(f), use the substitution
$ \omega= 2 \pi f $.
More specifically
$ \mathcal{X}(-\omega)=\mathcal{X}(-2\pi f)=X(-f) \ $
$ Since\ X(\alpha)=\mathcal{X}(2\pi \alpha) \ $
