The Inverse of a function with logic operators
The above is an injective function so that means it has an Inverse but will that be p= ¬q?


Yup. We have $f(T)=F, f(F)=T$ so $f^{-1}(F)=T, f^{-1}(T)=F$, which means that $f^{-1}(q)=\lnot q$. In this case $f^{-1}=f$: they're the same function.