average rate of change of f from a to b = [f(b) - f(a)] / (b - a).
f(x) = -1 / 3x
f(4) = -1 / 3(4) = -1 / 12
f(-4) = -1 / 3(-4) = -1 / -12 = 1 / 12
So,
ave = [f(b) - f(a)] / (b - a)
= [f(4) - f(-4)] / (4 - (-4))
= (-1/12 - 1/12) / 8
= (-1/6) / 8
= -1 / 48