Answer:
[tex](fg)(x) =8 \sqrt[6]{ (3 - x)^ {5 }} [/tex]
[tex](\frac{f}{g} )(x) = \frac{1}{2} \sqrt[6]{3 - x} [/tex]
[tex](fg)(2) = 8[/tex]
[tex](\frac{f}{g} )(2) =\frac{1}{2} [/tex]
The domain of (fg)(x) is all real numbers
The domain of (f/g)(x) is x>3
Step-by-step explanation:
The given functions are
[tex]f(x) = 2 \sqrt{3 - x} [/tex]
and
[tex]g(x) = 4 \sqrt[3]{3 - x} [/tex]
We want to to evaluate:
[tex](fg)(x) = f(x) \cdot \: g(x)[/tex]
[tex](fg)(x) =( 2 \sqrt{3 - x} )\times ( 4\sqrt[3]{3 - x} )[/tex]
We multiply to get:
[tex](fg)(x) =8(3 - x)^{ \frac{1}{2} } ( 3 - x)^{ \frac{1}{3} } [/tex]
[tex](fg)(x) =8(3 - x)^ {\frac{1}{2 } + \frac{1}{3}} [/tex]
[tex](fg)(x) =8(3 - x)^ {\frac{5}{6 } } [/tex]
[tex](fg)(x) =8 \sqrt[6]{ (3 - x)^ {5 }} [/tex]
The domain of (fg)(x) is all real numbers.
Similarly;
[tex] (\frac{f}{g} )(x) = \frac{f(x)}{g(x)} [/tex]
[tex](\frac{f}{g} )(x) = \frac{2 \sqrt{3 - x} }{4 \sqrt[3]{3 - x} } = \frac{1}{2} \sqrt[6]{3 - x} [/tex]
The domain is all real numbers greater than 3.
We need to evaluate (fg)(2) and (f/g)(2) to get:
[tex](fg)(2) =8 \sqrt[6]{ (3 - 2)^ {5 }} = 8 \sqrt[6]{ {1}^{5} } = 8[/tex]
[tex](\frac{f}{g} )(2) = \frac{1}{2} \sqrt[6]{3 - 2} = \frac{1}{2} [/tex]
