Given:
[tex]\( f(x) = \sqrt{2x} \)[/tex]
[tex]\( g(x) = \sqrt{18x} \)[/tex]

Find [tex]\((f \cdot g)(x)\)[/tex], assuming [tex]\(x \geq 0\)[/tex].

A. [tex]\((f \cdot g)(x) = 18x\)[/tex]
B. [tex]\((f \cdot g)(x) = 6x\)[/tex]
C. [tex]\((f \cdot g)(x) = \sqrt{20x}\)[/tex]
D. [tex]\((f \cdot g)(x) = 6\sqrt{x}\)[/tex]



Answer :

To find [tex]\((f \cdot g)(x)\)[/tex], where [tex]\(f(x) = \sqrt{2x}\)[/tex] and [tex]\(g(x) = \sqrt{18x}\)[/tex], we need to multiply these two functions together.

Starting from the given functions:
[tex]\[ f(x) = \sqrt{2x} \][/tex]
[tex]\[ g(x) = \sqrt{18x} \][/tex]

The product of [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex] is:
[tex]\[ (f \cdot g)(x) = f(x) \cdot g(x) \][/tex]

Substituting the expressions for [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex]:
[tex]\[ (f \cdot g)(x) = \sqrt{2x} \cdot \sqrt{18x} \][/tex]

To multiply these two square roots together, we can combine the contents under a single square root:
[tex]\[ (f \cdot g)(x) = \sqrt{(2x) \cdot (18x)} \][/tex]

Multiplying the terms inside the square root:
[tex]\[ (f \cdot g)(x) = \sqrt{36x^2} \][/tex]

Next, simplify the square root:
[tex]\[ \sqrt{36x^2} = \sqrt{36} \cdot \sqrt{x^2} \][/tex]

Since [tex]\(\sqrt{36} = 6\)[/tex] and [tex]\(\sqrt{x^2} = x\)[/tex] for [tex]\(x \geq 0\)[/tex]:
[tex]\[ \sqrt{36x^2} = 6x \][/tex]

Thus, we have:
[tex]\[ (f \cdot g)(x) = 6x \][/tex]

Therefore, the correct answer is:
[tex]\[ \boxed{6x} \][/tex]