To compare the average rates of change of the two functions \( p(x) = \sqrt{x + 4} \) (the square root function) and \( f(x) = \sqrt[3]{x + 4} \) (the cube root function) over the interval \([-4, -3]\), we need to calculate the average rate of change for each function.
The average rate of change of a function \( g(x) \) over the interval \([a, b]\) is given by:
\[ \text{Average Rate of Change} = \frac{g(b) - g(a)}{b - a} \]
For the Square Root Function \( p(x) \):
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Calculate \( p(-4) \): \[ p(-4) = \sqrt{-4 + 4} = \sqrt{0} = 0 \]
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Calculate \( p(-3) \): \[ p(-3) = \sqrt{-3 + 4} = \sqrt{1} = 1 \]
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Find the average rate of change: \[ \text{Average Rate of Change} = \frac{p(-3) - p(-4)}{-3 - (-4)} = \frac{1 - 0}{-3 + 4} = \frac{1}{1} = 1 \]
For the Cube Root Function \( f(x) \):
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Calculate \( f(-4) \): \[ f(-4) = \sqrt[3]{-4 + 4} = \sqrt[3]{0} = 0 \]
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Calculate \( f(-3) \): \[ f(-3) = \sqrt[3]{-3 + 4} = \sqrt[3]{1} = 1 \]
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Find the average rate of change: \[ \text{Average Rate of Change} = \frac{f(-3) - f(-4)}{-3 - (-4)} = \frac{1 - 0}{-3 + 4} = \frac{1}{1} = 1 \]
Conclusion
Both functions have the same average rate of change of 1 over the interval \([-4, -3]\).
Thus, the response is: The functions have the same average rate of change of 1.