To find the inverse function of \( f(x) = \frac{5x - 1}{x + 2} \), follow these steps:

### 1. Express \( y = f(x) \):
\[
y = \frac{5x - 1}{x + 2}
\]

### 2. Swap \( y \) and \( x \):
\[
x = \frac{5y - 1}{y + 2}
\]

### 3. Solve for \( y \):
Multiply both sides by \( y + 2 \) to eliminate the denominator:
\[
x(y + 2) = 5y - 1
\]
Expand and rearrange to group terms involving \( y \):
\[
xy + 2x = 5y - 1
\]
Move all terms involving \( y \) to one side:
\[
xy - 5y = -2x - 1
\]
Factor out \( y \):
\[
y(x - 5) = -2x - 1
\]
Finally, solve for \( y \):
\[
y = \frac{-2x - 1}{x - 5}
\]

### 4. State the inverse function:
\[
f^{-1}(x) = \frac{-2x - 1}{x - 5}
\]

Would you like a deeper explanation, or do you have any questions?

Here are some related questions:
1. How can you verify if a function and its inverse are correct?
2. What is the domain of the inverse function \( f^{-1}(x) = \frac{-2x - 1}{x - 5} \)?
3. How does the horizontal line test relate to finding inverse functions?
4. What is the significance of the inverse function in real-world applications?
5. What happens if the original function is not one-to-one?

**Tip:** Always check if a function is one-to-one (injective) before attempting to find its inverse.

Learn how to find the inverse function of f(x) = (5x - 1) / (x + 2) step-by-step with detailed explanations. Understand the concept of inverse functions and algebraic manipulation involved in solving for the inverse. Suitable for high school students and anyone interested in algebra.

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