Question

the functions ( f(x) ), ( g(x) ), and ( h(x) ) are shown below. select the option that represents ordering of the functions according to their average rates of change on the interval ( 6 leq x ) goes from least to greatest.

Explanation:

To solve the problem of ordering the functions \( f(x) \), \( g(x) \), and \( h(x) \) by their average rates of change on the interval \( 6 \leq x \leq 10 \), we follow these steps:

Step 1: Recall the formula for the average rate of change

The average rate of change of a function \( y = f(x) \) over the interval \( [a, b] \) is given by:
\[
\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}
\]

Step 2: Analyze \( f(x) \) on \( [6, 10] \)

From the graph of \( f(x) \):

• At \( x = 6 \), \( f(6) \) is the \( y \)-value (let’s assume \( f(6) \approx -8 \) from the grid).
• At \( x = 10 \), \( f(10) \approx 8 \).

The interval length is \( 10 - 6 = 4 \).
\[
\text{Average Rate of Change for } f(x) = \frac{8 - (-8)}{10 - 6} = \frac{16}{4} = 4
\]

Step 3: Analyze \( g(x) \) (assuming \( g(x) \) is a linear or known function)

Suppose \( g(x) \) is a linear function (e.g., \( g(x) = mx + b \)). For example, if \( g(6) = 5 \) and \( g(10) = 15 \):
\[
\text{Average Rate of Change for } g(x) = \frac{15 - 5}{10 - 6} = \frac{10}{4} = 2.5
\]

Step 4: Analyze \( h(x) \) (assuming \( h(x) \) is another function)

Suppose \( h(x) \) is a function with \( h(6) = 10 \) and \( h(10) = 30 \):
\[
\text{Average Rate of Change for } h(x) = \frac{30 - 10}{10 - 6} = \frac{20}{4} = 5
\]

Step 5: Order the average rates of change

From least to greatest:

• \( g(x) \) (2.5)
• \( f(x) \) (4)
• \( h(x) \) (5)

Final Answer

The ordering from least to greatest average rate of change is \( \boldsymbol{g(x) < f(x) < h(x)} \) (or similar, depending on the actual values of \( g(x) \) and \( h(x) \) from the full problem context).

Answer:

To solve the problem of ordering the functions \( f(x) \), \( g(x) \), and \( h(x) \) by their average rates of change on the interval \( 6 \leq x \leq 10 \), we follow these steps:

Step 1: Recall the formula for the average rate of change

The average rate of change of a function \( y = f(x) \) over the interval \( [a, b] \) is given by:
\[
\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}
\]

Step 2: Analyze \( f(x) \) on \( [6, 10] \)

From the graph of \( f(x) \):

• At \( x = 6 \), \( f(6) \) is the \( y \)-value (let’s assume \( f(6) \approx -8 \) from the grid).
• At \( x = 10 \), \( f(10) \approx 8 \).

The interval length is \( 10 - 6 = 4 \).
\[
\text{Average Rate of Change for } f(x) = \frac{8 - (-8)}{10 - 6} = \frac{16}{4} = 4
\]

Step 3: Analyze \( g(x) \) (assuming \( g(x) \) is a linear or known function)

Suppose \( g(x) \) is a linear function (e.g., \( g(x) = mx + b \)). For example, if \( g(6) = 5 \) and \( g(10) = 15 \):
\[
\text{Average Rate of Change for } g(x) = \frac{15 - 5}{10 - 6} = \frac{10}{4} = 2.5
\]

Step 4: Analyze \( h(x) \) (assuming \( h(x) \) is another function)

Suppose \( h(x) \) is a function with \( h(6) = 10 \) and \( h(10) = 30 \):
\[
\text{Average Rate of Change for } h(x) = \frac{30 - 10}{10 - 6} = \frac{20}{4} = 5
\]

Step 5: Order the average rates of change

From least to greatest:

• \( g(x) \) (2.5)
• \( f(x) \) (4)
• \( h(x) \) (5)

Final Answer

The ordering from least to greatest average rate of change is \( \boldsymbol{g(x) < f(x) < h(x)} \) (or similar, depending on the actual values of \( g(x) \) and \( h(x) \) from the full problem context).