We are given the function $f(x) = 4^{3x - 1}$ and need to find the value of $f(2)$.

Step-by-step solution:

1. First, substitute $x = 2$ into the function: $f(2) = 4^{3(2) - 1}$

2. Simplify the exponent: $f(2) = 4^{6 - 1} = 4^5$

3. Now, calculate $4^5$: $4^5 = 4 \times 4 \times 4 \times 4 \times 4 = 1024$

Thus, $f(2) = 1024$.

Since no rounding is needed, the value of $f(2)$ is $\boxed{1024}$.

Let me know if you need further clarification! Here are 5 related questions to explore:

1. What is $f(3)$ for the same function?
2. How would the result change if the base were 3 instead of 4?
3. Can you find the inverse function of $f(x)$?
4. What is the derivative of $f(x)$ with respect to $x$?
5. If $f(x) = 1024$, what is the value of $x$?

Tip: When solving functions involving exponents, make sure to simplify the exponent first to make the calculations easier!