Question

Use the attached image to answer the question. 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 over the interval: [-4,-3]
A. The functions have the same average rate of change of 1.
B. The functions have the same average rate of change of 0.
C. The square root function has the greater average rate of change.
D. The cube root function has the greater average rate of change.

Answer 1

To compare the average rates of change for the functions \( p(x) = \sqrt{x+4} \) and \( f(x) = \sqrt[3]{x+4} \) over the interval \([-4, -3]\), you can use the formula for the average rate of change:

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

For the interval \([-4, -3]\):

1. **Calculate for \( p(x) \)**:

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

2. **Calculate for \( f(x) \)**:

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

### Conclusion:
Both functions \( p(x) \) and \( f(x) \) have the same average rate of change of 1 over the interval \([-4, -3]\).

The correct answer is:
**A. The functions have the same average rate of change of 1.**