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A curve labeled p of x and five points on the curve are plotted on a coordinate plane. The curve begins on the x-axis sloping upward from left to right, passes through quadrant 2, and then rises through quadrant 1.

Compare the average rates of change for the graphed square root function p(x)=x+4−−−−−√ and the cube root function f(x)=3^√x+4−−−−−3
over the interval [−4,−3]
.

(1 point)
Responses

The functions have the same average rate of change of 1.
The functions have the same average rate of change of 1.

The square root function has the greater average rate of change.
The square root function has the greater average rate of change.

The cube root function has the greater average rate of change.
The cube root function has the greater average rate of change.

The functions have the same average rate of change of 0.
The functions have the same average rate of change of 0.
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1 answer

To compare 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 need to calculate the average rates of change for both functions over that interval.

The average rate of change of a function \( g(x) \) over an interval \([a, b]\) is given by:

\[ \text{Average Rate of Change} = \frac{g(b) - g(a)}{b - a} \]

For \( p(x) = \sqrt{x + 4} \):

  1. Evaluate \( p(-4) \): \[ p(-4) = \sqrt{-4 + 4} = \sqrt{0} = 0 \]

  2. Evaluate \( p(-3) \): \[ p(-3) = \sqrt{-3 + 4} = \sqrt{1} = 1 \]

  3. Calculate the average rate of change: \[ \text{Average Rate of Change} = \frac{p(-3) - p(-4)}{-3 - (-4)} = \frac{1 - 0}{-3 + 4} = \frac{1}{1} = 1 \]

For \( f(x) = \sqrt[3]{x + 4} \):

  1. Evaluate \( f(-4) \): \[ f(-4) = \sqrt[3]{-4 + 4} = \sqrt[3]{0} = 0 \]

  2. Evaluate \( f(-3) \): \[ f(-3) = \sqrt[3]{-3 + 4} = \sqrt[3]{1} = 1 \]

  3. Calculate the average rate of change: \[ \text{Average Rate of Change} = \frac{f(-3) - f(-4)}{-3 - (-4)} = \frac{1 - 0}{-3 + 4} = \frac{1}{1} = 1 \]

Conclusion:

Both functions have the same average rate of change of \( 1 \) over the interval \([-4, -3]\). Therefore, the correct response is:

The functions have the same average rate of change of 1.