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

Step 1: Replace $f(x)$ with $y$

Let $y = \frac{2x}{1 - x}$.

Step 2: Solve for $x$ in terms of $y$

Multiply both sides by $(1 - x)$ to eliminate the denominator:

$y(1 - x) = 2x$

Distribute the $y$ on the left-hand side:

$y - yx = 2x$

Move all terms involving $x$ to one side:

$y = 2x + yx$

Factor out $x$ on the right-hand side:

$y = x(2 + y)$

Now solve for $x$:

$x = \frac{y}{2 + y}$

Step 3: Write the inverse function

The inverse of the function is:

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

Would you like more details, or do you have any questions?

Here are five related questions to explore:

1. How do you verify that two functions are inverses of each other?
2. What is the domain of the original function $f(x) = \frac{2x}{1 - x}$?
3. What is the domain and range of the inverse function $f^{-1}(x) = \frac{x}{2 + x}$?
4. How does the graph of a function compare to its inverse?
5. Can you find the inverse of $f(x) = \frac{3x + 1}{2 - 4x}$?

Tip: To verify if two functions are inverses, compose them: $f(f^{-1}(x))$ and $f^{-1}(f(x))$ should both equal $x$.