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Thanks for clarifying the function
Part A.
The average rate of change is the change in y over the interval divided by the change in x over the interval. The change in x is just the length of the interval.
So section A, the average from 1 to 2 is
Section B, the average from 3 to 4 is
Part B.
Section B is changing (32/8=) four times faster than section A.
f is an exponential function that not only increases its value but increases the rate at which it increases its value. Each step of one unit in x doubles the rate of increase. Here we stepped two units, from [1,2] to [3,4], so quadrupled the rate.