Let's break down the question into manageable parts to understand the behavior of each function and determine their average rates of change over the interval $2 \leq x \leq 5$.

Given functions:

1. $f(x) = x$

   • A linear function with a slope of 1. Its average rate of change over any interval is constant.

2. $g(x) = -4x + 50$

   • A linear function with a slope of $-4$. The rate of change is constant and equal to the slope, which is $-4$.

3. $h(x) = -x^2 + 7x - 4$

   • A quadratic function. The rate of change is not constant, but we can calculate its average rate of change over the interval $2 \leq x \leq 5$ by evaluating the difference in function values at the endpoints and dividing by the interval length.

Steps to calculate the average rates of change:

1. For $f(x) = x$:

   • The average rate of change from $x = 2$ to $x = 5$ is: $\text{Average rate of change of } f(x) = \frac{f(5) - f(2)}{5 - 2} = \frac{5 - 2}{3} = \frac{3}{3} = 1$

2. For $g(x) = -4x + 50$:

   • The average rate of change from $x = 2$ to $x = 5$ is: [ \text{Average rate of change of } g(x) = \frac{g(5) - g(2)}{5 - 2} = \frac{[-4(5) + 50] - [-4(2) + 50]}{5 - 2} ] [ = \frac{[-20 + 50] - [-8 + 50]}{3} = \frac{30 - 42}{3} = \frac{-12}{3} = -4 ]

3. For $h(x) = -x^2 + 7x - 4$:

   • The average rate of change from $x = 2$ to $x = 5$ is: $\text{Average rate of change of } h(x) = \frac{h(5) - h(2)}{5 - 2}$ First, evaluate $h(5)$ and $h(2)$: $h(5) = -(5)^2 + 7(5) - 4 = -25 + 35 - 4 = 6$ $h(2) = -(2)^2 + 7(2) - 4 = -4 + 14 - 4 = 6$ $\text{Average rate of change of } h(x) = \frac{6 - 6}{3} = \frac{0}{3} = 0$

Summary of average rates of change:

• $f(x)$: Average rate of change = $1$
• $g(x)$: Average rate of change = $-4$
• $h(x)$: Average rate of change = $0$

Ordering from least to greatest:

• $g(x)$ has the least average rate of change ($-4$).
• $h(x)$ has the next average rate of change ($0$).
• $f(x)$ has the greatest average rate of change ($1$).

So, the ordering from least to greatest is: $g(x), h(x), f(x)$.

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Would you like a more detailed explanation or help with another part of the problem? Here are some related questions to explore further:

1. What would the average rate of change be for these functions over a different interval, say $1 \leq x \leq 4$?
2. How would the shape of the graph influence the average rate of change of a function?
3. What is the significance of the slope of a linear function in terms of its rate of change?
4. How can you calculate instantaneous rates of change for these functions?
5. What happens to the average rate of change of quadratic functions when you choose different intervals?

Tip: Always remember that the average rate of change is essentially the slope of the secant line connecting two points on the graph of the function. For non-linear functions, it changes depending on the interval chosen.