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 $[-4, 5]$.