To determine the correct comparison between the estimated average rate of change for the cubic function \( b(x) = 3^{\sqrt{3x + 9}} \) and the square root function \( d(x) = -\sqrt{3x + 9} \) over the interval \([-12, -3]\), we can analyze the behavior of both functions within that interval.
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Function \( b(x) \):
- The function \( b(x) = 3^{\sqrt{3x + 9}} \) begins in quadrant 3 and is increasing through quadrants 2 and 1.
- Over the interval \([-12, -3]\), because it is increasing, it suggests that \( b(x) \) has a positive rate of change.
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Function \( d(x) \):
- The function \( d(x) = -\sqrt{3x + 9} \) is a decreasing function, as noted by the negative sign in front of the square root.
- This function will have a negative average rate of change over the interval \([-12, -3]\) since it is decreasing.
From these observations, we can conclude:
- The estimated average rate of change of \( b(x) \) is positive (since \( b(x) \) is increasing).
- The estimated average rate of change of \( d(x) \) is negative (since \( d(x) \) is decreasing).
With this in mind, the correct statement is:
The estimated average rate of change of \( b(x) \) is greater than the estimated average rate of change of \( d(x) \) because \( b(x) \) is increasing over the interval but \( d(x) \) is decreasing.
Thus, the appropriate response would be:
- The estimated average rate of change of b(x) is greater than the estimated average rate of change of d(x) because b(x) is increasing over the interval but d(x) is decreasing.