To determine which square root function has a smaller estimated average rate of change over the interval \([0, 9]\) than a cube root function, we first need to understand the average rate of change for a function over an interval \([a, b]\).
The average rate of change of a function \(f(x)\) from \(x = a\) to \(x = b\) is calculated using the formula:
\[ \text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a} \]
Let's calculate the average rate of change for each of the given square root functions over the interval \([0, 9]\) and compare them to the average rate of change of the cube root function.
Assumed cube root function: The general form of a cube root can be \(g(x) = x^{1/3}\).
Calculating this for the interval \( [0, 9] \): \[ g(9) = 9^{1/3} = 2.0 \quad \text{and} \quad g(0) = 0^{1/3} = 0 \] Thus, \[ \text{Average Rate of Change of } g(x) = \frac{g(9) - g(0)}{9 - 0} = \frac{2 - 0}{9} = \frac{2}{9} \approx 0.2222 \]
Now let's evaluate the average rate of change for each square root function.
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\(f(x)=\sqrt{3x} - 1\) \[ f(9) = \sqrt{27} - 1 \approx 5.196 - 1 = 4.196 \quad \text{and} \quad f(0) = \sqrt{0} - 1 = -1 \] \[ \text{Average Rate of Change} = \frac{4.196 - (-1)}{9 - 0} = \frac{5.196}{9} \approx 0.5773 \]
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\(f(x) = \sqrt{x} + 1\) \[ f(9) = \sqrt{9} + 1 = 3 + 1 = 4 \quad \text{and} \quad f(0) = \sqrt{0} + 1 = 1 \] \[ \text{Average Rate of Change} = \frac{4 - 1}{9 - 0} = \frac{3}{9} = \frac{1}{3} \approx 0.3333 \]
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\(f(x) = \sqrt{x} - 1\) \[ f(9) = \sqrt{9} - 1 = 3 - 1 = 2 \quad \text{and} \quad f(0) = \sqrt{0} - 1 = -1 \] \[ \text{Average Rate of Change} = \frac{2 - (-1)}{9 - 0} = \frac{3}{9} = \frac{1}{3} \approx 0.3333 \]
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\(f(x) = 13\sqrt{x} - 1\) \[ f(9) = 13 \cdot 3 - 1 = 39 - 1 = 38 \quad \text{and} \quad f(0) = 13 \cdot 0 - 1 = -1 \] \[ \text{Average Rate of Change} = \frac{38 - (-1)}{9 - 0} = \frac{39}{9} \approx 4.3333 \]
After calculating the average rates of change:
- \(f(x) = \sqrt{3x} - 1\): \(\approx 0.5773\)
- \(f(x) = \sqrt{x} + 1\): \(\approx 0.3333\)
- \(f(x) = \sqrt{x} - 1\): \(\approx 0.3333\)
- \(f(x) = 13\sqrt{x} - 1\): \(\approx 4.3333\)
The average rate of change for the cube root function is \(\approx 0.2222\).
The functions \(f(x) = \sqrt{x} + 1\) and \(f(x) = \sqrt{x} - 1\) both have an average rate of change greater than the cube root function, but the function \(f(x) = \sqrt{3x} - 1\) also has a greater average rate of change.
Thus, the only valid choice that has a smaller estimated average rate change than the cube root function is:
\(f(x) = \sqrt{x} - 1\).
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