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To analyze the given problem, we need to evaluate the average rates of change of the functions \( f(x) \), \( g(x) \), and \( h(x) \) over the interval \([-4 \leq x \leq 5]\).

### Steps to Solve:

1. **Function \( f(x) \) (Graph Analysis):**
   - We will use the graph of \( f(x) \) to estimate the average rate of change over the interval.
   
2. **Function \( g(x) \) (Tabular Data Analysis):**
   - Given table of values for \( g(x) \).
   - We can use the data points to calculate the average rate of change over the interval.

3. **Function \( h(x) \) (Equation Analysis):**
   - Given \( h(x) = x^2 - 5x + 42 \).
   - We can use calculus to find the average rate of change over the interval.

### Detailed Calculations:

#### 1. Function \( f(x) \):

Using the graph, identify the values of \( f(x) \) at \( x = -4 \) and \( x = 5 \).

Let's approximate:
- \( f(-4) \approx -2 \)
- \( f(5) \approx 5 \)

The average rate of change for \( f(x) \) over \([-4, 5]\) is:
\[ \text{Average rate of change} = \frac{f(5) - f(-4)}{5 - (-4)} = \frac{5 - (-2)}{5 + 4} = \frac{5 + 2}{9} = \frac{7}{9} \approx 0.78 \]

#### 2. Function \( g(x) \):

Using the tabular data, calculate the average rate of change over \([-4, 5]\).

\[
\begin{array}{c|c}
x & g(x) \\
\hline
-4 & 98 \\
-1 & 50 \\
2 & 20 \\
5 & 8 \\
8 & 14 \\
11 & 38 \\
\end{array}
\]

We will use the values at \( x = -4 \) and \( x = 5 \):

\[ \text{Average rate of change} = \frac{g(5) - g(-4)}{5 - (-4)} = \frac{8 - 98}{5 + 4} = \frac{-90}{9} = -10 \]

#### 3. Function \( h(x) \):

For the function \( h(x) = x^2 - 5x + 42 \), calculate \( h(-4) \) and \( h(5) \):

\[ h(-4) = (-4)^2 - 5(-4) + 42 = 16 + 20 + 42 = 78 \]
\[ h(5) = (5)^2 - 5(5) + 42 = 25 - 25 + 42 = 42 \]

The average rate of change for \( h(x) \) over \([-4, 5]\) is:

\[ \text{Average rate of change} = \frac{h(5) - h(-4)}{5 - (-4)} = \frac{42 - 78}{5 + 4} = \frac{-36}{9} = -4 \]

### Ordering from Least to Greatest Average Rates of Change:
1. \( g(x) \) with rate \(-10\)
2. \( h(x) \) with rate \(-4\)
3. \( f(x) \) with rate \(\approx 0.78\)

Would you like more details or have any questions? 

Here are some related questions for further exploration:
1. What is the significance of the average rate of change for a function over an interval?
2. How can the average rate of change be used to understand the behavior of a function?
3. How does the average rate of change differ from the instantaneous rate of change?
4. Can you derive the equation of the tangent line to \( h(x) \) at \( x = 5 \)?
5. How would you find the average rate of change for a piecewise function?

**Tip:** When working with graphs, precise identification of points is crucial for accurate calculations. Using a ruler or software tools can help with better estimations.

The functions f(x), g(x), and h(x) are shown. Select the option that represents the ordering of the functions according to their average rates of change on the interval -4 ≤ x ≤ 5 from least to greatest.

This page explains how to determine the average rates of change of the functions f(x), g(x), and h(x) over the interval [-4, 5]. Explore graph analysis, tabular data, and quadratic functions to calculate the rates of change and order the functions from least to greatest.

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