$f(x) = f(x-1) + g(x)$
$\quad =f(x-1) + f(x-1) + g(x/2)$
$\quad = 2.f(x-1) + f(x/2 -1) + g(x/4)$
$\quad \vdots$
For simplicity I remove the second term in the expansion and then tries to get a lower bound for $f(x)$.
$f(x) > 2.f(x-1)$
$\implies f(x) >2.2.f(x-2)$
$\vdots$
$\implies f(x) > 2^{x}f(1)$
$\implies f(x) > 2^x$
So, option B is true, exponential growth for $f(x).$