Answer::
Average rate of change(A(x)) of f(x) over the interval [a, b] is given by:

As per the statement:
Given :
f(x) = x+4 over the interval [4, 5]
At x = 4
f(4) = 8
at x = 5
f(5) = 9
Using the above formula: we have;

Similarly for :
f(x) = 3x-7 over the interval [-3, 7]
at x = -3
f(-3) = -16
At x = 7:
f(7) = 14
then;

For:
f(x) = 9x + 1 over the interval [-5, -2]
At x = -5
f(-5) = -44
At x = -2
f(-2) = -17
then;

For:
f(x) = 2x + 9 over the interval [1, 2]
At x = 1
f(1) = 11
At x =2
f(2)= 13
then;

We have to Arrange these functions from the least to the greatest value based on the average rate of change in the specified interval.

⇒
⇒x+4 < 2x + 9 < 3x-7 < 9x + 1
Therefore, the least to the greatest value based on the average rate of change in the specified interval is:
f(x) = x+4, f(x) = 2x+9, f(x) = 3x-7, f(x) = 9x+1,