To find the inverse of the function \( g(x) = \frac{2x}{3x - 1} \), we need to follow these steps:

### Step 1: Replace \( g(x) \) with \( y \)
\[
y = \frac{2x}{3x - 1}
\]

### Step 2: Swap \( x \) and \( y \)
To find the inverse, interchange \( x \) and \( y \):
\[
x = \frac{2y}{3y - 1}
\]

### Step 3: Solve for \( y \)
Now, solve this equation for \( y \):

1. Multiply both sides by \( 3y - 1 \) to eliminate the fraction:
\[
x(3y - 1) = 2y
\]

2. Distribute \( x \) on the left-hand side:
\[
3xy - x = 2y
\]

3. Get all the terms involving \( y \) on one side:
\[
3xy - 2y = x
\]

4. Factor out \( y \) from the left-hand side:
\[
y(3x - 2) = x
\]

5. Finally, solve for \( y \):
\[
y = \frac{x}{3x - 2}
\]

### Step 4: Write the inverse function
The inverse function is:
\[
g^{-1}(x) = \frac{x}{3x - 2}
\]

Would you like more details or have any questions?

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Here are 5 related questions:

1. How do you verify if two functions are inverses of each other?
2. What is the domain of the inverse function \( g^{-1}(x) = \frac{x}{3x - 2} \)?
3. Can the original function and its inverse ever intersect?
4. What are the horizontal and vertical asymptotes of the inverse function?
5. How does the graph of a function relate to the graph of its inverse?

**Tip:** To find the inverse of a function, always swap \( x \) and \( y \) first and then solve for \( y \). This approach ensures that you're correctly inverting the function.

Learn how to find the inverse of the function g(x) = 2x / (3x - 1) step-by-step with detailed explanations. This problem involves understanding inverse functions and their formulas. Suitable for advanced high school students.

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