Start with
$A^+=\{A\}$
Using
$A \to BC$,
we get
$A^+=\{A,B,C\}$
Now use
$C \to FG$
Therefore,
$A^+=\{A,B,C,F,G\}$
Using $G \to A$ adds nothing new.
There is no functional dependency that allows us to obtain $E$ or $H$ from these attributes.
Therefore,
$A^+=\{A,B,C,F,G\}$
So A is correct.
For $A \to E$ to belong to $F^+$, $E$ must belong to $A^+$.
But, $E \notin A^+$.
Therefore, B is incorrect.
For $A$ to be a candidate key,
$A^+$ must contain every attribute of $R$.
Since $E$ and $H$ are missing, $A$ is not even a superkey.
Therefore, C is incorrect.
Since
$B,F \in A^+$,
the functional dependency
$A \to BF$ is implied by $F$.
Therefore,
$A \to BF \in F^+$.
So D is correct.
