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We are tasked with finding the inverse of the function:

\[
f(x) = \frac{2x + 3}{x - 4}, \quad \text{where} \, x \neq 4
\]

### Steps to find the inverse function:

1. Start by replacing \( f(x) \) with \( y \):

   \[
   y = \frac{2x + 3}{x - 4}
   \]

2. Swap \( x \) and \( y \) to reflect the inverse operation:

   \[
   x = \frac{2y + 3}{y - 4}
   \]

3. Now, solve for \( y \) (the inverse function):
   - Multiply both sides by \( (y - 4) \) to eliminate the fraction:

   \[
   x(y - 4) = 2y + 3
   \]

   - Expand the left-hand side:

   \[
   xy - 4x = 2y + 3
   \]

   - Move all terms involving \( y \) to one side:

   \[
   xy - 2y = 4x + 3
   \]

   - Factor out \( y \) on the left-hand side:

   \[
   y(x - 2) = 4x + 3
   \]

   - Finally, solve for \( y \):

   \[
   y = \frac{4x + 3}{x - 2}
   \]

### Therefore, the inverse function is:

\[
f^{-1}(x) = \frac{4x + 3}{x - 2}, \quad \text{where} \, x \neq 2
\]

Would you like further clarification on any step, or do you have any other questions?

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Here are five related questions you might find useful:
1. How do we verify if the inverse function is correct?
2. What happens if \( x = 4 \) or \( x = 2 \) in the original and inverse functions, respectively?
3. How do the domains and ranges of the original and inverse functions relate?
4. What is the graphical interpretation of the inverse function?
5. Can we find the inverse of a quadratic function similarly?

**Tip:** To confirm an inverse function, substitute it into the original function, and you should retrieve \( x \) in both directions.

Determine the inverse of f(x) = (2x + 3) / (x - 4), where x ≠ 4.

Learn how to find the inverse of the rational function f(x) = (2x + 3) / (x - 4), where x ≠ 4, with a step-by-step solution. This example involves solving for y by swapping variables and manipulating equations. Ideal for students in Grades 11-12 studying inverse functions and rational equations.

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