It is not necessary to do that. A is an upper bound for 1/f(x) if and only if 1/A is a lower bound (not 0) for f(x) and B is a lower bound for 1/f(x) if and only if 1/B is an upper bound for f(x).
Find an upper and lower bound for the reciprocal of the modulus of if
Out of all the the ways to break modulus how do I know which structure of the triangle inequalities to use?
Why am I getting the same value for my upper and lower bounds?