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.

Question

Compare the average rates of change for the graphed square root function p(x)=x+4−−−−√ and the cube root function f(x)=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 functions have the same average rate of change of 0.
The functions have the same average rate of change of 0.

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

Answer 1

To compare the average rates of change of the functions \( p(x) = \sqrt{x + 4} \) (the square root function) and \( f(x) = \sqrt[3]{x + 4} \) (the 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. Average rate of change: \[ \text{Average Rate of Change for } p = \frac{p(-3) - p(-4)}{-3 - (-4)} = \frac{1 - 0}{-3 + 4} = \frac{1}{1} = 1 \]

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

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.