Question

consider the two functions f(x) = \sqrt{x} and g(x) = -\sqrt{x}.

1. what is the average rate of change for each function from x = 4 to x = 9?

b. how are the two values in part (a) related to each other?
c. suppose the average rate of change for f(x) between two values of x is 0.32. what is the average rate of change for g(x) between the same two values of x?
b. choose the correct answer below.
a. the values are opposites.
b. the values are reciprocals.
c. the values are equivalent.
d. the values are opposite reciprocals.
c. if the average rate of change for f(x) between two values of x is 0.32, then the average rate of change for g(x) between the same two values of x is \square.

Explanation:

Part b

To determine the relationship between the average rates of change of \( f(x)=\sqrt{x} \) and \( g(x) = -\sqrt{x} \), we recall the formula for the average rate of change of a function \( h(x) \) from \( x = a \) to \( x = b \): \( \text{Average Rate of Change}=\frac{h(b)-h(a)}{b - a} \).

For \( f(x)=\sqrt{x} \), the average rate of change from \( x = 4 \) to \( x = 9 \) is \( \frac{\sqrt{9}-\sqrt{4}}{9 - 4}=\frac{3 - 2}{5}=\frac{1}{5} = 0.2 \).

For \( g(x)=-\sqrt{x} \), the average rate of change from \( x = 4 \) to \( x = 9 \) is \( \frac{-\sqrt{9}-(-\sqrt{4})}{9 - 4}=\frac{- 3+2}{5}=\frac{-1}{5}=- 0.2 \).

We can see that the average rate of change of \( g(x) \) is the negative (opposite) of the average rate of change of \( f(x) \). So the two values (average rates of change) are opposites.

Step 1: Recall the relationship

From part (b), we know that the average rate of change of \( g(x) \) is the opposite of the average rate of change of \( f(x) \) between the same two values of \( x \).

Step 2: Apply the relationship

Given that the average rate of change for \( f(x) \) is \( 0.32 \), then the average rate of change for \( g(x) \) will be the negative of \( 0.32 \), i.e., \( - 0.32 \).

Answer:

A. The values are opposites.

Part c