Compare the average rates of change for the square root function p\left(x\right)=\sqrt{x+4} and the cube root function f\left(x\right)=\sqrt[3]{x+4} over the interval [-4, -3].
A-the square root function has the greater average rate of change.
B-the cube root function has the greater average rate of change.
C-the functions have the same average rate of change of 1.
D-the functions have the same average rate of change of 0.
To find the average rate of change of a function over an interval, we need to find the change in the function's value divided by the change in the input variable. In this case, the interval is \([-4, -3]\).
For the square root function, we have \(p(x) = \sqrt{x + 4}\). The value of the function at the endpoints of the interval is \(p(-4) = \sqrt{0} = 0\) and \(p(-3) = \sqrt{1} = 1\). The change in the function's value is \(1 - 0 = 1\), and the change in the input variable is \((-3) - (-4) = 1\). Therefore, the average rate of change of the square root function over the interval \([-4, -3]\) is \(1/1 = 1\).
For the cube root function, we have \(f(x) = \sqrt[3]{x + 4}\). The value of the function at the endpoints of the interval is \(f(-4) = \sqrt[3]{0} = 0\) and \(f(-3) = \sqrt[3]{1} = 1\). The change in the function's value is \(1 - 0 = 1\), and the change in the input variable is \((-3) - (-4) = 1\). Therefore, the average rate of change of the cube root function over the interval \([-4, -3]\) is \(1/1 = 1\).
Since both functions have the same average rate of change of 1 over the interval \([-4, -3]\), the correct answer is:
C) the functions have the same average rate of change of 1.