To find [tex]\((f \cdot g)(x)\)[/tex] given the functions [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex], we proceed as follows:
1. Write Down the Functions:
[tex]\(f(x) = \sqrt{6x}\)[/tex]
[tex]\(g(x) = \sqrt{24x}\)[/tex]
2. Define the Product Function [tex]\((f \cdot g)(x)\)[/tex]:
[tex]\((f \cdot g)(x) = f(x) \cdot g(x)\)[/tex]
Substituting the given functions:
[tex]\((f \cdot g)(x) = \sqrt{6x} \cdot \sqrt{24x}\)[/tex]
3. Combine the Square Roots:
Using the property of square roots, [tex]\(\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}\)[/tex]:
[tex]\[ \sqrt{6x} \cdot \sqrt{24x} = \sqrt{(6x) \cdot (24x)} \][/tex]
4. Multiply the Terms Inside the Square Root:
[tex]\[ (6x) \cdot (24x) = 6 \cdot 24 \cdot x \cdot x = 144x^2 \][/tex]
So,
[tex]\[ \sqrt{(6x) \cdot (24x)} = \sqrt{144x^2} \][/tex]
5. Simplify the Square Root:
We know that [tex]\(\sqrt{a^2} = a\)[/tex] for any non-negative [tex]\(a\)[/tex]:
[tex]\[ \sqrt{144x^2} = 12x \][/tex]
6. Conclude the Result:
Therefore, [tex]\((f \cdot g)(x) = 12x\)[/tex].
When we compare this result with the given options:
- A. [tex]\((f \cdot g)(x) = 12x\)[/tex] (This matches our result)
- B. [tex]\((f \cdot g)(x) = \sqrt{30x}\)[/tex]
- C. [tex]\((f \cdot g)(x) = 72x\)[/tex]
- D. [tex]\((f \cdot g)(x) = 12\sqrt{x}\)[/tex]
So, the correct answer is:
A. [tex]\((f \cdot g)(x) = 12x\)[/tex]
