For which of the following functions is the average rate of change from to equal to ?

To determine which function has an average rate of change equal to , we need to calculate the average rate of change for each function and compare the results.

The average rate of change of a function over an interval is given by the formula:

average rate of change = (f(b) - f(a)) / (b - a)

where f(b) represents the value of the function at the upper limit of the interval, f(a) represents the value of the function at the lower limit of the interval, and (b - a) represents the length of the interval.

Let's calculate the average rate of change for each function:

1. Function 1: f(x) = 2x + 3 average rate of change = (f(b) - f(a)) / (b - a) = (2b + 3 - (2a + 3)) / (b - a) = (2b - 2a) / (b - a) = 2

2. Function 2: f(x) = x^2 + 5 average rate of change = (f(b) - f(a)) / (b - a) = ((b^2 + 5) - (a^2 + 5)) / (b - a) = (b^2 - a^2) / (b - a) = (b + a)

3. Function 3: f(x) = 3x - 2 average rate of change = (f(b) - f(a)) / (b - a) = (3b - 2 - (3a - 2)) / (b - a) = (3b - 3a) / (b - a) = 3

Now, we compare the average rates of change for each function to :

• Function 1 has an average rate of change of 2.
• Function 2 has an average rate of change of (b + a).
• Function 3 has an average rate of change of 3.

Therefore, the function with an average rate of change equal to is Function 3: f(x) = 3x - 2.