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 \( x \).

Learn how to find the inverse of the function f(x) = 2x / (1 - x) in this detailed, step-by-step solution. This problem covers key algebraic concepts such as inverse functions and solving equations. Suitable for students in Grades 9-12.

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