The functional equation provided is:

$f(x + y) = f(x) \cdot f(y)$

And you are given:

$f(2) = 4$

The goal is to find $f(4)$.

Approach:

To solve this, observe that the functional equation resembles the form of an exponential function. Let’s assume:

$f(x) = a^x$

Substituting this into the functional equation $f(x + y) = f(x) \cdot f(y)$:

$a^{x + y} = a^x \cdot a^y$

This is valid for any base $a$, which confirms that the functional equation holds for exponential functions. Now, we use the given $f(2) = 4$ to find $a$:

$f(2) = a^2 = 4 \implies a = 2$

Thus, the function is:

$f(x) = 2^x$

Finally, to find $f(4)$:

$f(4) = 2^4 = 16$

Thus, the answer is B) 16.

Would you like a further breakdown of any steps or a more detailed explanation?

Here are some related questions:

1. How do exponential functions satisfy functional equations like this one?
2. What is the general solution to functional equations of this form?
3. How would the solution change if the functional equation were different (e.g., $f(x + y) = f(x) + f(y)$)?
4. What other types of functions could potentially satisfy the functional equation $f(x + y) = f(x) \cdot f(y)$?
5. What role do initial conditions (such as $f(2) = 4$) play in solving functional equations?

Tip: Exponential functions often come up in functional equations that exhibit multiplication behavior across sums.