g(6) = (1/3)(6) + 3 because the expression you stated for g(x) simply has one third as the coefficient, i.e. it's a linear function--if something else was intended, you would have to include parentheses to control the PEMDAS evaluation.
Therefore g(6) = 5
and consequently f(g(6)) = f(5) = 6(5) - 3 = 27