To find the product of the functions \( f(x) \) and \( g(x) \), we need to compute \( (f \cdot g)(x) = f(x) \cdot g(x) \).
Given the functions:
[tex]\[ f(x) = \sqrt{3x} \][/tex]
[tex]\[ g(x) = \sqrt{48x} \][/tex]
Let's determine \( (f \cdot g)(x) \):
Step 1: Write down the product \( f(x) \cdot g(x) \):
[tex]\[ (f \cdot g)(x) = \left(\sqrt{3x}\right) \cdot \left(\sqrt{48x}\right) \][/tex]
Step 2: Use the property of square roots \(\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}\):
[tex]\[ (f \cdot g)(x) = \sqrt{(3x) \cdot (48x)} \][/tex]
Step 3: Simplify the expression inside the square root:
[tex]\[ (f \cdot g)(x) = \sqrt{3x \cdot 48x} \][/tex]
[tex]\[ (f \cdot g)(x) = \sqrt{3 \cdot 48 \cdot x \cdot x} \][/tex]
[tex]\[ (f \cdot g)(x) = \sqrt{144x^2} \][/tex]
Step 4: Simplify the square root:
[tex]\[ \sqrt{144x^2} = \sqrt{144} \cdot \sqrt{x^2} \][/tex]
[tex]\[ \sqrt{144x^2} = 12 \cdot x \][/tex]
So, \( (f \cdot g)(x) = 12x \).
Thus, the correct answer is:
[tex]\[ \boxed{12x} \][/tex]
The answer is Option D:
[tex]\[ (f \cdot g)(x) = 12x \][/tex]
