Area under Functional Equation?

The required function above is $f(x) = 2^{x}$ , and the area region in question simplifies to :

$3|x| + 2|y| \le 8 \Rightarrow -4 + \frac{3}{2}|x| \le y \le 4 - \frac{3}{2}|x|$

which is a parallelogram that is centered about the origin and has vertices at $(x,y) = (0, \pm4); (\pm \frac{8}{3}, 0)$ in the $xy-$ plane. The total area is just the sum of four right triangles with leg lengths of 4 and 8/3 each:

$Area = 4\cdot\frac{1}{2}\cdot(\frac{8}{3})(4) = \frac{64}{3} = \frac{2^{6}}{3} = \frac{f(6)}{3}.$

Hence, $\boxed{n = 3}.$