Which is the inverse function of the function $$ f(x)=\frac{x}{3 x+2} $$

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To find the inverse function of $$ f(x) = \frac{x}{3x + 2} $$, we will follow these steps:

1. Replace $$ f(x) $$ with $$ y $$:
   $$
   y = \frac{x}{3x + 2}
   $$

2. Swap $$ x $$ and $$ y $$:
   $$
   x = \frac{y}{3y + 2}
   $$

3. Solve for $$ y $$:
   - First, multiply both sides by $$ 3y + 2 $$ to eliminate the fraction:
   $$
   x(3y + 2) = y
   $$
   - Distributing $$ x $$:
   $$
   3xy + 2x = y
   $$
   - Rearranging the equation to isolate terms involving $$ y $$:
   $$
   3xy - y = -2x
   $$
   - Factor out $$ y $$ on the left side:
   $$
   y(3x - 1) = -2x
   $$
   - Now, solve for $$ y $$:
   $$
   y = \frac{-2x}{3x - 1}
   $$

4. Thus, the inverse function is:
   $$
   f^{-1}(x) = \frac{-2x}{3x - 1}
   $$

To summarize, the inverse function of $$ f(x) = \frac{x}{3x + 2} $$ is:
$$
f^{-1}(x) = \frac{-2x}{3x - 1}
$$
The inverse function of $$ f(x) = \frac{x}{3x + 2} $$ is $$ f^{-1}(x) = \frac{-2x}{3x - 1} $$.

Which is the inverse function of the function \( f(x)=\frac{x}{3 x+2} \)

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