1-4 is correct.

For 5, you should mention that b has to be non-zero and without loss of generality b>0

f(a) = f((a/b)*b) = f((a/b) + (a/b) + (a/b) ... + (a/b)) [adding b times] = f(a/b)f(a/b)--f(a/b) [multiplying the function by itself b times] = [f(a/b)]^b

Hence f(a/b) = [f(a)]^(1/b) = (q^a)^(1/b) [by 4] = q^(a/b) [by exponent rules] which is what we expected.