To find \((f \cdot g)(x)\), we start by determining the product of the two functions \(f(x)\) and \(g(x)\).
Given: \[ f(x) = \frac{\sqrt{x + 3}}{x} \] \[ g(x) = \frac{\sqrt{x + 3}}{2x} \]
The product \( (f \cdot g)(x) \) can be calculated as follows: \[ (f \cdot g)(x) = f(x) \cdot g(x) = \left(\frac{\sqrt{x + 3}}{x}\right) \cdot \left(\frac{\sqrt{x + 3}}{2x}\right) \]
Now, we multiply the numerators and the denominators: \[ = \frac{\sqrt{x + 3} \cdot \sqrt{x + 3}}{x \cdot 2x} = \frac{x + 3}{2x^2} \]
So, we find that: \[ (f \cdot g)(x) = \frac{x + 3}{2x^2} \]
Therefore, the correct response is: \[ \boxed{\frac{x + 3}{2x^2}} \]