To find the average gradient (or average rate of change) of the function $g(x)$ between the points $A(-3, -5)$ and $B(-1, 52)$, we use the following formula for the average gradient between two points $(x_1, y_1)$ and $(x_2, y_2)$ on a function:

$\text{Average Gradient} = \frac{y_2 - y_1}{x_2 - x_1}$

Substituting the coordinates of points $A$ and $B$:

$\text{Average Gradient} = \frac{52 - (-5)}{-1 - (-3)}$

Now, simplify the expression:

$\text{Average Gradient} = \frac{52 + 5}{-1 + 3} = \frac{57}{2} = 28.5$

So, the average gradient of $g(x)$ between the points $A(-3, -5)$ and $B(-1, 52)$ is $28.5$.

Would you like more details, or do you have any questions?

Here are 5 related questions:

1. What is the gradient of the function $g(x)$ at a specific point, such as $x = -2$?
2. How would you find the equation of the tangent to the curve $g(x)$ at point $B$?
3. What is the significance of the gradient in the context of an exponential function?
4. How does the gradient change as $x$ increases for an exponential function?
5. How do you calculate the instantaneous rate of change for $g(x)$ at a particular point?

Tip: When calculating the average gradient, ensure that you correctly subtract the coordinates in the right order to avoid sign errors.