To find the average rates of change of the functions \( p(x) = \sqrt{x + 4} \) and \( f(x) = \sqrt[3]{x + 4} \) over the interval \([-4, -3]\), we will calculate the average rate of change for both functions using the formula:
\[ \text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a} \]
where \( a = -4 \) and \( b = -3 \).
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For the square root function \( p(x) = \sqrt{x + 4} \):
\[ p(-4) = \sqrt{-4 + 4} = \sqrt{0} = 0 \] \[ p(-3) = \sqrt{-3 + 4} = \sqrt{1} = 1 \]
Now calculate the average rate of change:
\[ \text{Average Rate of Change} = \frac{p(-3) - p(-4)}{-3 - (-4)} = \frac{1 - 0}{-3 + 4} = \frac{1}{1} = 1 \]
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For the cube root function \( f(x) = \sqrt[3]{x + 4} \):
\[ f(-4) = \sqrt[3]{-4 + 4} = \sqrt[3]{0} = 0 \] \[ f(-3) = \sqrt[3]{-3 + 4} = \sqrt[3]{1} = 1 \]
Now calculate the average rate of change:
\[ \text{Average Rate of Change} = \frac{f(-3) - f(-4)}{-3 - (-4)} = \frac{1 - 0}{-3 + 4} = \frac{1}{1} = 1 \]
Thus, for both functions, the average rates of change over the interval \([-4, -3]\) are equal to 1.
Based on this calculation, the correct response is:
The functions have the same average rate of change of 1.
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