Let $f:A\to B$ be a commutative ring morphism. Let $I\vartriangleleft A$ be an ideal. If $f(I)\subset \mathrm J(B)$ is contained in the Jacobson radical then $f(1+i)=1+f(i)\in B^\times$ is a unit so $1+I$ is inverted.
On the other hand, if $1+f(i)\in B^\times$ it does not seem to follow that $f(i)\in \mathrm J(B)$ since perhaps $1+bf(i)\notin B^\times$ for some $b\in B$.
Question. What are some examples where this implication fails i.e $f(1+I)\subset B^\times$ but $f(I)\nsubseteq\mathrm J(B)$?