To determine the average rates of change for the functions \( b(x) \) and \( d(x) \) over the interval \([-12, -3]\), we can use the formula for the average rate of change of a function \( f(x) \) over an interval \([a, b]\):
\[ \text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a} \]
For \( b(x) = \sqrt{3x + 9} \):
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Calculate \( b(-12) = \sqrt{3(-12) + 9} = \sqrt{-36 + 9} = \sqrt{-27} \) (not real, so it is undefined for this point, we can skip it).
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Calculate \( b(-3) = \sqrt{3(-3) + 9} = \sqrt{-9 + 9} = \sqrt{0} = 0 \).
Since \( b(x) \) is undefined at \( -12 \), we cannot get an average rate of change over the interval \([-12, -3]\) using the conventional method that needs both endpoints defined. However, \( b(x) \) has a valid output at \( -3 \).
For \( d(x) = \sqrt{-3x + 9} \):
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Calculate \( d(-12) = \sqrt{-3(-12) + 9} = \sqrt{36 + 9} = \sqrt{45} \).
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Calculate \( d(-3) = \sqrt{-3(-3) + 9} = \sqrt{9 + 9} = \sqrt{18} \).
Now we can compute the average rate of change for \( d(x) \):
\[ \text{Average Rate of Change of } d(x) = \frac{d(-3) - d(-12)}{-3 - (-12)} = \frac{\sqrt{18} - \sqrt{45}}{-3 + 12} = \frac{\sqrt{18} - \sqrt{45}}{9} \]
Comparing Average Rates of Change:
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Since \( b(x) \) is undefined at one of the endpoints of the interval, this complicates direct comparison unless we analyze the behavior of both functions.
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As \( x \) decreases from \(-3\) to more negative values, \( b(x) \) is increasing and moving towards non-reality (since it becomes undefined).
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For \( d(x) \), since we know movement is toward the negative \( x \), it holds values, and we note:
- Near the endpoints, \( d(x) \) is well-defined, moving from positive values down as it decreases.
Conclusion
Given the analysis, without strictly computed numbers but understanding the behavior:
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The estimated average rate of change of \( b(x) \) can't be assessed due to undefined behavior at the starting point \(-12\).
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The function \( d(x) \) has a positive output and thus illustrates an increasing behavior.
If forced on an interpretation on the responses:
- The third option notes that \( d(x) \) displays positive movement, reflecting broader trends, although \( b(x) \) ceases to be valid across the interval.
Therefore, the most fitting statement from the responses is:
The estimated average rate of change of \( d(x) \) is greater than the estimated average rate of change of \( b(x) \) because \( d(x) \) is positive over the interval but \( b(x) \) is negative (not valid).
Make sure to check that the curves are analyzed appropriately to confirm nature beyond simple numeric outcomes!