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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 functionf(x)=x+4−−−−√3
over the interval [−4,−3]
.

(1 point)
Responses

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.

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.
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1 answer

To compare the average rates of change for the functions \( p(x) = \sqrt{x + 4} \) (square root function) and \( f(x) = \sqrt[3]{x + 4} \) (cube root function) over the interval \([-4, -3]\), we need to calculate the average rate of change for both functions over that interval.

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

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

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

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

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

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

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

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

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

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

Conclusion:

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

Thus, the correct response is: The functions have the same average rate of change of 1.