To find the average rate of change of the function $f(x) = e^{x - 1} - 3$ between $x = 1$ and $x = 3$, we use the formula:

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

where $x_1 = 1$ and $x_2 = 3$.

1. Calculate $f(1)$:

   $f(1) = e^{1 - 1} - 3 = e^0 - 3 = 1 - 3 = -2$

2. Calculate $f(3)$:

   $f(3) = e^{3 - 1} - 3 = e^2 - 3$

   The value of $e^2$ is approximately $7.389$, so:

   $f(3) \approx 7.389 - 3 = 4.389$

3. Calculate the Average Rate of Change:

   $\text{Average Rate of Change} = \frac{f(3) - f(1)}{3 - 1} = \frac{4.389 - (-2)}{2} = \frac{4.389 + 2}{2} = \frac{6.389}{2} = 3.2$

So, the average rate of change is approximately 3.2.

Would you like further explanation or details?

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Here are some related questions:

1. How does the average rate of change differ from the instantaneous rate of change?
2. How can we use derivatives to find the rate of change at any given point?
3. What would the average rate of change be over a different interval, say from $x = 1$ to $x = 4$?
4. What are the applications of the average rate of change in real-world problems?
5. How does the exponential base $e$ affect the growth rate of exponential functions?

Tip: When finding average rates of change, make sure to evaluate the function accurately at each endpoint before calculating the difference.