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To solve this problem, we need to compute the average rates of change for each function \( f(x) \), \( g(x) \), and \( h(x) \) over the interval \( 1 \leq x \leq 3 \), and then order them from least to greatest.

### Step 1: Average Rate of Change Formula
The average rate of change of a function \( y = f(x) \) over an interval \([a, b]\) is given by:

\[
\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}
\]

We will apply this formula to each of the three functions over the interval \( [1, 3] \).

---

### Step 2: Finding Average Rate of Change for \( g(x) \)
From the table, we can see that:

- \( g(1) = 3 \)
- \( g(3) = 3 \)

Thus, the average rate of change for \( g(x) \) over \( [1, 3] \) is:

\[
\frac{g(3) - g(1)}{3 - 1} = \frac{3 - 3}{2} = 0
\]

---

### Step 3: Finding Average Rate of Change for \( h(x) \)
The function \( h(x) \) is given as \( h(x) = x^2 - 2x + 10 \). We calculate:

- \( h(1) = 1^2 - 2(1) + 10 = 9 \)
- \( h(3) = 3^2 - 2(3) + 10 = 13 \)

Thus, the average rate of change for \( h(x) \) over \( [1, 3] \) is:

\[
\frac{h(3) - h(1)}{3 - 1} = \frac{13 - 9}{2} = \frac{4}{2} = 2
\]

---

### Step 4: Finding Average Rate of Change for \( f(x) \)
Using the graph of \( f(x) \), estimate:

- \( f(1) \approx 4 \)
- \( f(3) \approx -4 \)

Thus, the average rate of change for \( f(x) \) over \( [1, 3] \) is:

\[
\frac{f(3) - f(1)}{3 - 1} = \frac{-4 - 4}{2} = \frac{-8}{2} = -4
\]

---

### Step 5: Ordering the Rates of Change
The average rates of change for the three functions are:

- \( f(x) \): -4
- \( g(x) \): 0
- \( h(x) \): 2

Thus, the ordering from least to greatest is:

\[
f(x) < g(x) < h(x)
\]

### Final Answer: The correct ordering of the functions according to their average rates of change is \( f(x) < g(x) < h(x) \).

Would you like more details or have any questions?

### Related Questions:
1. What does the average rate of change represent geometrically?
2. How do we interpret a rate of change of 0 for a function?
3. Can the average rate of change be negative and what does that signify?
4. How does the formula for the average rate of change relate to the slope of a line?
5. What is the difference between instantaneous rate of change and average rate of change?

### Tip:
When calculating the average rate of change from a graph, always carefully estimate the function values at the endpoints to avoid rounding errors.

Select the option that represents the ordering of the functions f(x), g(x), and h(x) according to their average rates of change on the interval 1 ≤ x ≤ 3 from least to greatest.

Learn how to determine the average rate of change for functions f(x), g(x), and h(x) over the interval 1 ≤ x ≤ 3. This problem involves understanding functions, intervals, and rate of change formulas, suitable for high school math students in grades 9-11.

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