To solve the given problem, we need to find [tex]\(\left(\frac{f}{g}\right)(x)\)[/tex] for the given functions [tex]\( f(x) = \sqrt[3]{3x} \)[/tex] and [tex]\( g(x) = 3x + 2 \)[/tex]. Additionally, we need to determine any restrictions on the domain.
Let's go through the solution step-by-step:
### Step 1: Define the Functions
Given:
[tex]\[ f(x) = \sqrt[3]{3x} \][/tex]
[tex]\[ g(x) = 3x + 2 \][/tex]
### Step 2: Form the Function [tex]\(\left(\frac{f}{g}\right)(x)\)[/tex]
To find [tex]\(\left(\frac{f}{g}\right)(x)\)[/tex], we divide [tex]\( f(x) \)[/tex] by [tex]\( g(x) \)[/tex]:
[tex]\[
\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)} = \frac{\sqrt[3]{3x}}{3x + 2}
\][/tex]
### Step 3: Determine Restrictions on the Domain
For the fraction [tex]\(\frac{\sqrt[3]{3x}}{3x + 2}\)[/tex] to be defined, the denominator [tex]\( g(x) \)[/tex] must not be zero. Therefore, we set:
[tex]\[
g(x) \neq 0
\][/tex]
[tex]\[
3x + 2 \neq 0
\][/tex]
Now, solve for [tex]\( x \)[/tex]:
[tex]\[
3x + 2 = 0
\][/tex]
[tex]\[
3x = -2
\][/tex]
[tex]\[
x = -\frac{2}{3}
\][/tex]
So, [tex]\( x \)[/tex] must not be [tex]\(-\frac{2}{3}\)[/tex].
### Conclusion
Putting it all together:
[tex]\[
\left(\frac{f}{g}\right)(x) = \frac{\sqrt[3]{3x}}{3x + 2}
\][/tex]
with the restriction [tex]\( x \neq -\frac{2}{3} \)[/tex].
Thus, the correct answer is:
[tex]\[
\boxed{\text{A. } \left(\frac{f}{g}\right)(x)=\frac{\sqrt[3]{3 x}}{3 x+2}, x \neq -\frac{2}{3}}
\][/tex]
