Question

Can you please help me on me review?

Answer 1

Answer:

$(fg)(x) =8 \sqrt[6]{ (3 - x)^ {5 }}$

$(\frac{f}{g} )(x) = \frac{1}{2} \sqrt[6]{3 - x}$

$(fg)(2) = 8$

$(\frac{f}{g} )(2) =\frac{1}{2}$

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

$f(x) = 2 \sqrt{3 - x}$

and

$g(x) = 4 \sqrt[3]{3 - x}$

We want to to evaluate:

$(fg)(x) = f(x) \cdot \: g(x)$

$(fg)(x) =( 2 \sqrt{3 - x} )\times ( 4\sqrt[3]{3 - x} )$

We multiply to get:

$(fg)(x) =8(3 - x)^{ \frac{1}{2} } ( 3 - x)^{ \frac{1}{3} }$

$(fg)(x) =8(3 - x)^ {\frac{1}{2 } + \frac{1}{3}}$

$(fg)(x) =8(3 - x)^ {\frac{5}{6 } }$

$(fg)(x) =8 \sqrt[6]{ (3 - x)^ {5 }}$

The domain of (fg)(x) is all real numbers.

Similarly;

$(\frac{f}{g} )(x) = \frac{f(x)}{g(x)}$

$(\frac{f}{g} )(x) = \frac{2 \sqrt{3 - x} }{4 \sqrt[3]{3 - x} } = \frac{1}{2} \sqrt[6]{3 - x}$

The domain is all real numbers greater than 3.

We need to evaluate (fg)(2) and (f/g)(2) to get:

$(fg)(2) =8 \sqrt[6]{ (3 - 2)^ {5 }} = 8 \sqrt[6]{ {1}^{5} } = 8$

$(\frac{f}{g} )(2) = \frac{1}{2} \sqrt[6]{3 - 2} = \frac{1}{2}$