Let's solve the given problem step by step:

Part (a) - Average Rate of Change:

The function given is $g(x) = 2 - 5x$.

The average rate of change of a function between two points $(x_1, y_1)$ and $(x_2, y_2)$ is calculated using the formula:

$\text{Average Rate of Change} = \frac{g(x_2) - g(x_1)}{x_2 - x_1}$

Here, the points given are $(-3, 17)$ and $(3, -13)$. Let's verify these values:

• For $x = -3$: $g(-3) = 2 - 5(-3) = 2 + 15 = 17$ This matches the point $(-3, 17)$.

• For $x = 3$: $g(3) = 2 - 5(3) = 2 - 15 = -13$ This matches the point $(3, -13)$.

Now, we calculate the average rate of change: $\frac{g(3) - g(-3)}{3 - (-3)} = \frac{-13 - 17}{3 + 3} = \frac{-30}{6} = -5$

So, the average rate of change is -5.

Part (b) - Function Behavior:

The function $g(x) = 2 - 5x$ is a linear function with a slope of $-5$. Since the slope is constant and negative, the function is decreasing on the interval $-3 \leq x \leq 3$.

Would you like any further details?

Here are 5 related questions:

1. What is the average rate of change of $g(x) = 2 - 5x$ between $x = 0$ and $x = 3$?
2. How does the slope of a linear function affect its graph?
3. What is the equation of the line tangent to the graph of $g(x)$ at $x = 0$?
4. How do you find the average rate of change of a quadratic function?
5. What is the difference between average rate of change and instantaneous rate of change?

Tip: The slope of a linear function is constant, meaning the average rate of change will always be the same, no matter which points you choose.