We are given the exponential function $f(x) = 2 \cdot 3^x$, and we are asked to evaluate $f(2) - f(1)$.

Step 1: Calculate $f(2)$

Using the given function: $f(2) = 2 \cdot 3^2 = 2 \cdot 9 = 18$

Step 2: Calculate $f(1)$

Similarly: $f(1) = 2 \cdot 3^1 = 2 \cdot 3 = 6$

Step 3: Subtract $f(1)$ from $f(2)$

$f(2) - f(1) = 18 - 6 = 12$

So, the value of $f(2) - f(1)$ is $\boxed{12}$.

Would you like further details or have any questions?

Here are some related questions:

1. What is the general form of an exponential function?
2. How does the base of the exponent affect the growth of the function?
3. What happens when the base of an exponential function is less than 1?
4. Can you evaluate $f(0)$ for this function?
5. How would the graph of $f(x) = 2 \cdot 3^x$ look?

Tip: In exponential functions, the base of the exponent determines whether the function models growth (if the base is greater than 1) or decay (if the base is between 0 and 1).