Sure, let's find the solution using some simple steps.
**Step 1: Function Definition**
The first step is to start with your function, which is f(x) = x^2 + 3x + 5.
**Step 2: Function Evaluation**
In order to find the average rate of change, we need to substitute x = 4 and x = 5 into the equation:
So, substituting x = 4 into the function, we get:
f(4) = (4)^2 + 3*(4) + 5 = 16 + 12 + 5 = 33
Similarly, substituting x = 5 into the function, we get:
f(5) = (5)^2 + 3*(5) + 5 = 25 + 15 + 5 = 45
So, the function values at x = 4 and x = 5 are 33 and 45 respectively.
**Step 3: Average Rate of Change**
Now that we have the function values, the average rate of change of the function on the interval [4,5] is given by:
((f(x2) - f(x1)) / (x2 - x1))
Substituting the values into the above equation gives:
((f(5) - f(4)) / (5 - 4)) = (45 - 33) / (5 - 4) = 12
So, the average rate of change of the function on the interval [4,5] is 12. And that's your answer.