To solve for [tex]\((f \cdot g)(x)\)[/tex] given the functions [tex]\(f(x) = \sqrt{2x}\)[/tex] and [tex]\(g(x) = \sqrt{32x}\)[/tex], we proceed as follows:
1. Expression for [tex]\((f \cdot g)(x)\)[/tex]:
The notation [tex]\((f \cdot g)(x)\)[/tex] represents the product of [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex]. So, we need to multiply the expressions of [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex]:
[tex]\[
(f \cdot g)(x) = f(x) \cdot g(x)
\][/tex]
2. Substitute the expressions for [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex]:
Recall that [tex]\(f(x) = \sqrt{2x}\)[/tex] and [tex]\(g(x) = \sqrt{32x}\)[/tex]. Substituting these into the product:
[tex]\[
(f \cdot g)(x) = \sqrt{2x} \cdot \sqrt{32x}
\][/tex]
3. Simplify the expression:
Using the property of square roots [tex]\(\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}\)[/tex], we combine the square roots:
[tex]\[
(f \cdot g)(x) = \sqrt{2x \cdot 32x}
\][/tex]
4. Multiply the terms inside the square root:
Simplify [tex]\(2x \cdot 32x\)[/tex]:
[tex]\[
2x \cdot 32x = 64x^2
\][/tex]
5. Take the square root of the resulting expression:
[tex]\[
\sqrt{64x^2} = \sqrt{64} \cdot \sqrt{x^2} = 8 \cdot x
\][/tex]
Thus, the product function [tex]\((f \cdot g)(x)\)[/tex] simplifies to:
[tex]\[
(f \cdot g)(x) = 8x
\][/tex]
To see this in action, let's plug in a specific value of [tex]\(x\)[/tex]:
- Let [tex]\(x = 4\)[/tex]:
[tex]\[
f(4) = \sqrt{2 \cdot 4} = \sqrt{8} \approx 2.828
\][/tex]
[tex]\[
g(4) = \sqrt{32 \cdot 4} = \sqrt{128} \approx 11.314
\][/tex]
[tex]\[
(f \cdot g)(4) = 2.828 \cdot 11.314 \approx 32
\][/tex]
Therefore, for [tex]\(x = 4\)[/tex], [tex]\((f \cdot g)(x) \approx 32.000\)[/tex].
The final result matches the calculated result for [tex]\(x = 4\)[/tex], verifying our derived function:
[tex]\[
(f \cdot g)(x) = 8x
\][/tex]
