The function given is $f(x) = 6 \cdot 3^x$, which is an exponential function.

To determine whether the average rate of change of this function is constant or variable, let's analyze its nature.

Average Rate of Change:

The average rate of change of a function $f(x)$ over an interval $[a, b]$ is given by:

$\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}$

For exponential functions like $f(x) = 6 \cdot 3^x$, the rate of change depends on the values of $a$ and $b$ chosen. This is because exponential functions grow by a constant factor (in this case, by a factor of 3) rather than by a constant amount.

Conclusion:

The average rate of change of $f(x) = 6 \cdot 3^x$ varies and is not constant. This is a characteristic of exponential functions, which do not have a constant rate of change like linear functions.

Would you like more details on how the rate of change varies for this function?

Here are 5 related questions:

1. How do you calculate the average rate of change for any function?
2. What is the difference between linear and exponential functions in terms of growth?
3. How do you determine the instantaneous rate of change for exponential functions?
4. Can exponential functions model real-world population growth effectively?
5. What is the impact of the base (3 in this case) on the growth rate of the function?

Tip: For exponential growth, the base of the exponent (like 3 here) indicates the multiplication factor for each unit increase in the input.