Show that if f(x) and g(x) are functions from the set of real numbers to the set of real numbers, then f(x) is ? (g (x)) if and only if there are positive constants k, C1, and C2 such that C1|g(x)| ? |f(x)| ? C2|g(x)| whenever x > k.

Solution Step 1:f(x) and are the functions from set of real numbers to the set of real numbers,then we have to prove is if and only if there are positive constants k,and such that .Step 2:First we will prove is implies there are positive constants k,and such that and there are positive constants k,and such that implies is Consider is then there are some constants and such that for all …(1)And for all for all (multiply both sides by ..(2)From (1) and (2) we get …(3)Thus we proved is implies there are positive constants k,and such that …(4)