To find the product [tex]\( (f \cdot g)(x) \)[/tex], we start with the given functions:
[tex]\[ f(x) = \frac{\sqrt{x+3}}{x} \][/tex]
[tex]\[ g(x) = \frac{\sqrt{x+3}}{2x} \][/tex]
The product of [tex]\( f \)[/tex] and [tex]\( g \)[/tex], denoted by [tex]\( (f \cdot g)(x) \)[/tex], is given by:
[tex]\[ (f \cdot g)(x) = f(x) \cdot g(x). \][/tex]
Now, substitute the given functions into the expression:
[tex]\[ (f \cdot g)(x) = \left( \frac{\sqrt{x+3}}{x} \right) \cdot \left( \frac{\sqrt{x+3}}{2x} \right). \][/tex]
To multiply these fractions together, we multiply the numerators and the denominators separately:
[tex]\[ (f \cdot g)(x) = \frac{\sqrt{x+3} \cdot \sqrt{x+3}}{x \cdot 2x}. \][/tex]
Simplify the expression inside the numerator and the denominator:
[tex]\[ \sqrt{x+3} \cdot \sqrt{x+3} = (\sqrt{x+3})^2 = x+3, \][/tex]
[tex]\[ x \cdot 2x = 2x^2. \][/tex]
Thus, the product becomes:
[tex]\[ (f \cdot g)(x) = \frac{x+3}{2x^2}. \][/tex]
So, the final expression for [tex]\( (f \cdot g)(x) \)[/tex] is:
[tex]\[
(f \cdot g)(x) = \frac{x+3}{2x^2}.
\][/tex]
