Explanation:
To find the average rate of change of the function f(x) over the interval [-2,4], we need to calculate the change in the function value divided by the change in the input variable over that interval:
Average rate of change = (f(4) - f(-2))/(4 - (-2))
First, we calculate f(4):
f(4) = √(4 - 2) = √2
Next, we calculate f(-2):
f(-2) = √(-2 - 2) = √(-4)
Since the square root of a negative number is not a real number, the function f(x) is not defined for x < 2. Therefore, the interval [-2,4] is not entirely within the domain of the function.
However, we can still find the average rate of change over the part of the interval that is within the domain of the function, which is [2,4]. Therefore, we need to modify the formula accordingly:
Average rate of change = (f(4) - f(2))/(4 - 2)
f(2) = √(2 - 2) = 0
Plugging in the values we get:
Average rate of change = (√2 - 0)/(4 - 2) ≈ 0.71
Therefore, the average rate of change of f(x) over the interval [-2,4] (within the domain of the function) to the nearest hundredth is 0.71.