[tex]\[
\begin{array}{l}
f(x)=\sqrt{3x} \\
g(x)=\sqrt{48x}
\end{array}
\][/tex]

Find \((f \cdot g)(x)\). Assume \(x \geq 0\).

A. \((f \cdot g)(x)=\sqrt{51x}\)
B. \((f \cdot g)(x)=72x\)
C. \((f \cdot g)(x)=12\sqrt{x}\)
D. [tex]\((f \cdot g)(x)=12x\)[/tex]



Answer :

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]