To solve this problem, let’s start by finding the product of the functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex]. Given:
[tex]\[ f(x) = x - 6 \][/tex]
[tex]\[ g(x) = x^{\frac{1}{2}}(x + 3) \][/tex]
First, we simplify [tex]\( g(x) \)[/tex]:
[tex]\[ g(x) = x^{\frac{1}{2}} \cdot (x + 3) \][/tex]
[tex]\[ g(x) = x^{\frac{1}{2}} \cdot x + x^{\frac{1}{2}} \cdot 3 \][/tex]
[tex]\[ g(x) = x^{1 + \frac{1}{2}} + 3x^{\frac{1}{2}} \][/tex]
[tex]\[ g(x) = x^{\frac{3}{2}} + 3x^{\frac{1}{2}} \][/tex]
Next, we find the product [tex]\( g(x) \times f(x) \)[/tex]:
[tex]\[ g(x) \times f(x) = (x^{\frac{3}{2}} + 3x^{\frac{1}{2}}) \times (x - 6) \][/tex]
We use the distributive property to simplify this expression:
[tex]\[ g(x) \times f(x) = x^{\frac{3}{2}} \cdot x + x^{\frac{3}{2}} \cdot (-6) + 3x^{\frac{1}{2}} \cdot x + 3x^{\frac{1}{2}} \cdot (-6) \][/tex]
[tex]\[ g(x) \times f(x) = x^{\frac{5}{2}} - 6x^{\frac{3}{2}} + 3x^{\frac{3}{2}} - 18x^{\frac{1}{2}} \][/tex]
[tex]\[ g(x) \times f(x) = x^{\frac{5}{2}} - 3x^{\frac{3}{2}} - 18x^{\frac{1}{2}} \][/tex]
To express the terms with exponents in a common form, note that:
[tex]\[ x^{\frac{5}{2}} = \sqrt{x^5} \][/tex]
[tex]\[ x^{\frac{3}{2}} = \sqrt{x^3} \][/tex]
[tex]\[ x^{\frac{1}{2}} = \sqrt{x} \][/tex]
Thus, the expression simplifies to:
[tex]\[ g(x) \times f(x) = \sqrt{x^5} - 3\sqrt{x^3} - 18\sqrt{x} \][/tex]
The correct answer is:
B. [tex]\(\sqrt{x^5} - 3\sqrt{x^3} - 18\sqrt{x}\)[/tex]
