To determine [tex]\((f \cdot g)(x)\)[/tex] where [tex]\( f(x) = \sqrt{3x} \)[/tex] and [tex]\( g(x) = \sqrt{48x} \)[/tex], let's follow these steps:
1. Write the product function:
We are asked to find [tex]\( (f \cdot g)(x) \)[/tex], which means multiplying [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex]:
[tex]\[
(f \cdot g)(x) = f(x) \cdot g(x) = \sqrt{3x} \cdot \sqrt{48x}
\][/tex]
2. Utilize properties of square roots:
Recall that the product of square roots can be combined under a single square root:
[tex]\[
\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}
\][/tex]
3. Combine under one square root:
Apply this property to our product:
[tex]\[
(f \cdot g)(x) = \sqrt{3x} \cdot \sqrt{48x} = \sqrt{(3x) \cdot (48x)}
\][/tex]
4. Simplify the expression inside the square root:
Multiply the constants and [tex]\(x\)[/tex]:
[tex]\[
(3x) \cdot (48x) = 144x^2
\][/tex]
5. Simplify the square root:
Evaluate the square root of [tex]\( 144x^2 \)[/tex]:
[tex]\[
\sqrt{144x^2} = \sqrt{144} \cdot \sqrt{x^2} = 12 \cdot x = 12x
\][/tex]
Therefore, [tex]\((f \cdot g)(x) = 12x\)[/tex].
So, the correct answer is:
[tex]\[
\boxed{12x}
\][/tex]
