Let's go through the steps needed to find \((f \cdot g)(x)\) given the functions:
[tex]\[ f(x) = \sqrt{3x} \][/tex]
[tex]\[ g(x) = \sqrt{48x} \][/tex]
To find \((f \cdot g)(x)\), we need to determine the product of \(f(x)\) and \(g(x)\):
[tex]\[ (f \cdot g)(x) = f(x) \cdot g(x) \][/tex]
First, let's express \(f(x)\) and \(g(x)\) in detail:
[tex]\[ f(x) = \sqrt{3x} \][/tex]
[tex]\[ g(x) = \sqrt{48x} \][/tex]
To find the product, we multiply these two expressions together:
[tex]\[ (f \cdot g)(x) = \sqrt{3x} \cdot \sqrt{48x} \][/tex]
Recall the property of square roots:
[tex]\[ \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} \][/tex]
Applying this property to our functions:
[tex]\[ (f \cdot g)(x) = \sqrt{3x} \cdot \sqrt{48x} = \sqrt{(3x) \cdot (48x)} \][/tex]
Now multiply the expressions inside the square root:
[tex]\[ (3x) \cdot (48x) = 3 \cdot 48 \cdot x \cdot x = 144x^2 \][/tex]
So, the expression becomes:
[tex]\[ (f \cdot g)(x) = \sqrt{144x^2} \][/tex]
Since \(\sqrt{x^2} = |x|\) and we are given that \(x \geq 0\):
[tex]\[ \sqrt{144x^2} = 12x \][/tex]
Thus, the function \((f \cdot g)(x)\) is:
[tex]\[ (f \cdot g)(x) = 12x \][/tex]
Therefore, the correct answer is:
D. [tex]\((f \cdot g)(x) = 12x\)[/tex]
