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To find the inverse of the given function \( f(x) = \frac{1x - 6}{2x + 4} \), follow these steps:

### Step-by-Step Solution:

1. **Rewrite the function:** Set \( y = f(x) \) so that:
   \[
   y = \frac{x - 6}{2x + 4}
   \]

2. **Swap \(x\) and \(y\):** To find the inverse, interchange \(x\) and \(y\):
   \[
   x = \frac{y - 6}{2y + 4}
   \]

3. **Solve for \(y\):** 
   - Multiply both sides by \(2y + 4\) to eliminate the denominator:
     \[
     x(2y + 4) = y - 6
     \]
   - Distribute \(x\) on the left-hand side:
     \[
     2xy + 4x = y - 6
     \]
   - Group all terms involving \(y\) on one side:
     \[
     2xy - y = -4x - 6
     \]
   - Factor out \(y\) on the left-hand side:
     \[
     y(2x - 1) = -4x - 6
     \]
   - Solve for \(y\):
     \[
     y = \frac{-4x - 6}{2x - 1}
     \]

Thus, the inverse function is:
\[
f^{-1}(x) = \frac{-4x - 6}{2x - 1}
\]

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

### Related questions:
1. How do you verify that two functions are inverses of each other?
2. What is the domain of the inverse function \( f^{-1}(x) \)?
3. How do you find the derivative of an inverse function?
4. Can a function have more than one inverse?
5. What happens if a function is not one-to-one? Can it still have an inverse?

**Tip:** When solving for the inverse of a function, always remember to swap \(x\) and \(y\) and then solve for \(y\) to get the inverse function.

Given f(x) = (x - 6) / (2x + 4), find the inverse function f^(-1)(x).

Learn how to find the inverse of the rational function f(x) = (x - 6) / (2x + 4) with a detailed step-by-step solution. This problem covers key algebraic concepts like inverse functions and solving rational expressions. Suitable for high school students in Grades 10-12.

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