Question

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

a. 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 )?

1. the average rate of change from ( x = 4 ) to ( x = 9 ) of ( f(x) ) is ( \frac{1}{5} ).

(simplify your answer)

the average rate of change from ( x = 4 ) to ( x = 9 ) of ( g(x) ) is (\boxed{quad}).

Explanation:

Step1: Recall the average rate of change formula

The average rate of change of a function \( y = h(x) \) from \( x = a \) to \( x = b \) is given by \( \frac{h(b)-h(a)}{b - a} \).

For \( g(x)=-\sqrt{x} \), we need to find the average rate of change from \( x = 4 \) to \( x = 9 \). First, find \( g(4) \) and \( g(9) \).

Step2: Calculate \( g(4) \) and \( g(9) \)

• For \( x = 4 \): \( g(4)=-\sqrt{4}=- 2 \)
• For \( x = 9 \): \( g(9)=-\sqrt{9}=-3 \)

Step3: Apply the average rate of change formula

Using the formula \( \frac{g(9)-g(4)}{9 - 4} \), substitute the values:
\( \frac{-3-(-2)}{9 - 4}=\frac{-3 + 2}{5}=\frac{-1}{5} \)

Answer:

\( -\frac{1}{5} \)