To find the inverse of the function \( f(x) = \frac{x+4}{-x+1} \), follow these steps:
1. **Replace \( f(x) \) with \( y \):**
\( y = \frac{x+4}{-x+1} \)
2. **Swap \( x \) and \( y \):**
Interchange \( x \) and \( y \): \( x = \frac{y+4}{-y+1} \)
3. **Solve for \( y \):**
Multiply both sides by \(-y+1\): \( x(-y+1) = y+4 \)
Distribute \( x \): \( -xy+x = y+4 \)
Subtract \( y \) from both sides: \( -xy+x-y = 4 \)
Factor out \( -y \) from the left side: \( -y(x-1) = 4-x \)
Divide by \( -(x-1) \): \( y = \frac{x-4}{x-1} \)
So, the inverse function \( f^{-1}(x) \) is \( f^{-1}(x) = \frac{x-4}{x-1} \).