Let's start by analyzing the given functions:
1. The function \( f(x) \) is defined as:
[tex]\[
f(x) = \sqrt{2x}
\][/tex]
2. The function \( g(x) \) is defined as:
[tex]\[
g(x) = \sqrt{50x}
\][/tex]
We want to find the product of these two functions, denoted as \( (f \cdot g)(x) \). This means we need to multiply \( f(x) \) and \( g(x) \) together. Let's write this down:
[tex]\[
(f \cdot g)(x) = f(x) \cdot g(x) = \sqrt{2x} \cdot \sqrt{50x}
\][/tex]
Next, we use the property of square roots that \( \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} \) to combine the square roots:
[tex]\[
(f \cdot g)(x) = \sqrt{(2x) \cdot (50x)} = \sqrt{100x^2}
\][/tex]
Simplify the expression inside the square root:
[tex]\[
(f \cdot g)(x) = \sqrt{100x^2}
\][/tex]
Since the square root of \( 100x^2 \) is \( 10x \) (considering \( x \geq 0 \)):
[tex]\[
(f \cdot g)(x) = 10x
\][/tex]
For an example calculation, let's take \( x = 1 \):
1. Calculate \( f(1) \):
[tex]\[
f(1) = \sqrt{2 \cdot 1} = \sqrt{2} \approx 1.4142135623730951
\][/tex]
2. Calculate \( g(1) \):
[tex]\[
g(1) = \sqrt{50 \cdot 1} = \sqrt{50} \approx 7.0710678118654755
\][/tex]
3. Finally, calculate \( (f \cdot g)(1) \):
[tex]\[
(f \cdot g)(1) = 10 \cdot 1 = 10.000000000000002
\][/tex]
Thus, the values for \( x = 1 \) are:
[tex]\[
f(1) \approx 1.4142135623730951, \quad g(1) \approx 7.0710678118654755, \quad (f \cdot g)(1) \approx 10.000000000000002
\][/tex]
In summary, the product function [tex]\( (f \cdot g)(x) \)[/tex] simplifies to [tex]\( 10x \)[/tex], and we verified this with [tex]\( x = 1 \)[/tex] giving the detailed numerical results as [tex]\( f(1) \approx 1.4142135623730951 \)[/tex], [tex]\( g(1) \approx 7.0710678118654755 \)[/tex], and [tex]\( (f \cdot g)(1) \approx 10.000000000000002 \)[/tex].
