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

Answer 1

To compare the average rates of change of the two 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 need to calculate the average rate of change for each function.

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

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

### For the Square Root Function \( p(x) \):

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. Find 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 the Cube Root Function \( f(x) \):

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