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F(x)=(x+4)/(-x+1), find the inverse

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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 \( f^{-1}(x) = \frac{x-4}{x-1} \).
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