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.