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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 square root function has the greater average rate of change.
The square 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.
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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 can use the formula for the average rate of change:

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

where \( a = -4 \) and \( b = -3 \).

For the square root function \( p(x) = \sqrt{x + 4} \):

  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 for \( p(x) \): \[ \text{Average Rate of Change for } p(x) = \frac{p(-3) - p(-4)}{-3 - (-4)} = \frac{1 - 0}{-3 + 4} = \frac{1}{1} = 1 \]

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

  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 for \( f(x) \): \[ \text{Average Rate of Change for } f(x) = \frac{f(-3) - f(-4)}{-3 - (-4)} = \frac{1 - 0}{-3 + 4} = \frac{1}{1} = 1 \]

Conclusion:

Both the square root function \( p(x) \) and the cube root function \( f(x) \) have the same average rate of change of 1 over the interval \([-4, -3]\).

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