To find the inverse of the function \( y = \frac{x}{x^2 - 4} \), we need to swap \( x \) and \( y \), and then solve for \( y \). Let's walk through the process:

1. Start with the equation:
   \[
   y = \frac{x}{x^2 - 4}
   \]
   
2. Swap \( x \) and \( y \) to set up for finding the inverse:
   \[
   x = \frac{y}{y^2 - 4}
   \]

3. Multiply both sides by \( y^2 - 4 \) to get rid of the fraction:
   \[
   x(y^2 - 4) = y
   \]

4. Expand the left-hand side:
   \[
   xy^2 - 4x = y
   \]

5. Rearrange the equation to isolate terms involving \( y \):
   \[
   xy^2 - y = 4x
   \]

6. Factor out \( y \) on the left side:
   \[
   y(xy - 1) = 4x
   \]

7. Solve for \( y \):
   \[
   y = \frac{4x}{xy - 1}
   \]

Thus, the inverse function is \( y = \frac{4x}{xy - 1} \). However, note that this may not always be the simplest form of the inverse function, and further simplifications might be attempted depending on specific conditions.

Would you like a more detailed explanation of these steps or any further clarifications?

Here are five related questions:
1. How can you verify if two functions are inverses of each other?
2. What conditions need to be checked to ensure a function has an inverse?
3. What role do domain and range play when finding an inverse function?
4. Can you find the inverse of a function graphically?
5. How do asymptotes affect the inverse of a rational function?

**Tip:** Always check the domain of the original function when finding its inverse to avoid including values that make the function undefined (like division by zero).

Discover how to find the inverse of the rational function y = x / (x^2 - 4). This problem explores inverse functions and rational expressions, with a step-by-step solution. Suitable for students in Grades 11-12 studying advanced algebra.

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