The problem asks for the average rate of change of the function $g(x) = 1x^3 + 3$ over the interval $x = -2$ to $x = 1$.

Formula for Average Rate of Change:

The average rate of change of a function $g(x)$ over an interval $[a, b]$ is given by: $\text{Average rate of change} = \frac{g(b) - g(a)}{b - a}$

Step-by-Step Solution:

1. The function is $g(x) = x^3 + 3$.
2. The interval is from $x = -2$ to $x = 1$.
   • Here, $a = -2$ and $b = 1$.

Find $g(1)$:

$g(1) = (1)^3 + 3 = 1 + 3 = 4$

Find $g(-2)$:

$g(-2) = (-2)^3 + 3 = -8 + 3 = -5$

Calculate the Average Rate of Change:

$\frac{g(1) - g(-2)}{1 - (-2)} = \frac{4 - (-5)}{1 + 2} = \frac{4 + 5}{3} = \frac{9}{3} = 3$

Final Answer:

The average rate of change of $g(x)$ from $x = -2$ to $x = 1$ is 3.

Would you like more details or have any questions?

Here are 5 related questions:

1. How do you find the average rate of change for a different interval?
2. What is the difference between average rate of change and instantaneous rate of change?
3. Can you find the rate of change for other polynomial functions, such as quadratic functions?
4. How does the average rate of change relate to the slope of a secant line?
5. What happens to the average rate of change if the function was a linear equation instead?

Tip: The average rate of change gives you the slope of the secant line between two points on a curve!