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]
