To solve for [tex]\((f \cdot g)(x)\)[/tex] where [tex]\( f(x) = \sqrt{32x} \)[/tex] and [tex]\( g(x) = \sqrt{2x} \)[/tex], follow these steps:
### Step-by-Step Solution:
1. Understand the Functions:
- We have [tex]\( f(x) = \sqrt{32x} \)[/tex].
- We also have [tex]\( g(x) = \sqrt{2x} \)[/tex].
2. Define the Product of the Functions:
- The product of two functions, [tex]\( (f \cdot g)(x) \)[/tex], is calculated as [tex]\( f(x) \cdot g(x) \)[/tex].
3. Set Up the Expression:
- [tex]\( (f \cdot g)(x) = \sqrt{32x} \cdot \sqrt{2x} \)[/tex].
4. Combine the Square Roots:
- Recall the property of square roots: [tex]\( \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} \)[/tex]:
[tex]\[
(f \cdot g)(x) = \sqrt{32x} \cdot \sqrt{2x} = \sqrt{(32x) \cdot (2x)}.
\][/tex]
5. Simplify the Expression:
- Simplify inside the square root:
[tex]\[
(32x) \cdot (2x) = 64x^2.
\][/tex]
- Therefore:
[tex]\[
(f \cdot g)(x) = \sqrt{64x^2}.
\][/tex]
6. Evaluate the Square Root:
- Simplify the square root of [tex]\( 64x^2 \)[/tex]:
[tex]\[
\sqrt{64x^2} = \sqrt{64} \cdot \sqrt{x^2} = 8 \cdot x = 8x.
\][/tex]
### Conclusion:
The function that describes [tex]\((f \cdot g)(x)\)[/tex] is:
[tex]\[
(f \cdot g)(x) = 8x.
\][/tex]
Thus, the correct answer is:
[tex]\[
\boxed{8x}
\][/tex]
Therefore, the correct option is C. [tex]\((f \cdot g)(x)=8x\)[/tex].
