Question

consider the two functions ( f(x) = sqrt{x} ) and ( g(x) = -sqrt{x} ). 2. what is the average rate of change for each function from ( x = 4 ) to ( x = 9 )? a. the average rate of change from ( x = 4 ) to ( x = 9 ) of ( f(x) ) is (\boxed{quad}). (simplify your answer.) 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 )?

Explanation:

Step1: Recall the formula for average rate of change

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

Step2: Identify \( a \), \( b \), \( f(a) \), and \( f(b) \) for \( f(x)=\sqrt{x} \)

Here, \( a = 4 \), \( b = 9 \), \( f(4)=\sqrt{4}=2 \), and \( f(9)=\sqrt{9}=3 \).

Step3: Calculate the average rate of change

Substitute these values into the formula: \( \frac{f(9)-f(4)}{9 - 4}=\frac{3 - 2}{5}=\frac{1}{5}=0.2 \).

Answer:

\( \frac{1}{5} \) (or \( 0.2 \))