Here Option A is Wrong Cz if fn is 1/n
f(n)==1/2=.5
f(n)^2 =.25
.25 < .5
Option B is Wrong CZ
Take f(n)=2n & g(n)=n Which satisfies Antecedent part But not consequence part
Cz (2^2n=4^n )!=2^n
Now Why Option C is Correct.... [Answered By Will Win]
Let F(n) = n.
Let H(n) = O(F(n)).
That is, the order of growth of function H is asymptotically and inclusively upper bounded by the function F(n).
In simple words, H(n) <= F(n).
F(n) + O(F(n)) = Big Theta(F(n)) is correct as one can write is as
'n + order of growth of a function less than and EQUAL to F(n) = Asymptotic growth of F(n)'.
Note: O(F(n)) will not be asymptomatically faster than F(n).
Because H(n) = O(F(n)) says it all.
While the asymptotic upper bound for F(n) isn't mentioned in option (c). And that was the tricky part.
Hence C is is the Correct Ans.