To find \((f \cdot g)(x)\), we first need to compute the product of the functions \(f(x)\) and \(g(x)\).
Given: \[ f(x) = \frac{\sqrt{x + 3}}{x} \] \[ g(x) = \frac{\sqrt{x + 3}}{2x} \]
Now, we calculate \((f \cdot g)(x)\): \[ (f \cdot g)(x) = f(x) \cdot g(x) = \left(\frac{\sqrt{x + 3}}{x}\right) \cdot \left(\frac{\sqrt{x + 3}}{2x}\right) \]
To multiply these two fractions, we multiply the numerators and the denominators: \[ (f \cdot g)(x) = \frac{\sqrt{x + 3} \cdot \sqrt{x + 3}}{x \cdot 2x} = \frac{(\sqrt{x + 3})^2}{2x^2} \]
Since \((\sqrt{x + 3})^2 = x + 3\), we can simplify this further: \[ (f \cdot g)(x) = \frac{x + 3}{2x^2} \]
Finally, this gives us: \[ (f \cdot g)(x) = \frac{x + 3}{2x^2} \]
Thus, the final answer is: \[ \boxed{\frac{x + 3}{2 x^2}} \]
Match this with the options given, and you will find that it corresponds to the third option: \[ (f \cdot g)(x) \equiv \frac{x + 3}{2 x^2} \]