To find the inverse function $f^{-1}(x)$ of the function $f(x) = \frac{2}{3x + 1}$, follow these steps:

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

2. Interchange $x$ and $y$:
   Since we want the inverse, we swap $x$ and $y$: $x = \frac{2}{3y + 1}$

3. Solve for $y$: Multiply both sides by $3y + 1$ to get rid of the denominator: $x(3y + 1) = 2$

   Expand this: $3xy + x = 2$

   Now, isolate $y$: $3xy = 2 - x$ $y = \frac{2 - x}{3x}$

4. Rewrite $y$ as $f^{-1}(x)$: $f^{-1}(x) = \frac{2 - x}{3x}$

Final Answer:

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

Would you like a step-by-step explanation of any part of this solution?

Here are five related questions you might find helpful:

1. How can we verify that two functions are inverses of each other?
2. What are the conditions for a function to have an inverse?
3. How does the inverse of a rational function affect its domain and range?
4. What is the importance of interchanging $x$ and $y$ when finding the inverse?
5. How do transformations (e.g., reflections) relate to finding inverses graphically?

Tip: When finding inverses, always start by writing $y = f(x)$ and then interchange $x$ and $y$ to solve for the inverse function.