Find \(f^{-1}(x)\) for \(f(x)=\frac{x}{x+1}\)

📝 Question

Let:

\[ f:\mathbb{R}\setminus\{-1\}\to \mathbb{R}\setminus\{1\}, \quad f(x)=\frac{x}{x+1} \]

Find \(f^{-1}(x)\).

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✅ Solution

🔹 Step 1: Let

\[ y=\frac{x}{x+1} \] —

🔹 Step 2: Solve for \(x\)

\[ y(x+1)=x \]

\[ yx+y=x \]

\[ y=x-xy \]

\[ y=x(1-y) \]

\[ x=\frac{y}{1-y} \] —

🔹 Step 3: Write inverse

Interchange \(x\) and \(y\):

:contentReference[oaicite:0]{index=0} —

🎯 Final Answer

\[ \boxed{f^{-1}(x)=\frac{x}{1-x}} \]

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🚀 Exam Shortcut

• Cross multiply first
• Collect terms of \(x\)
• Factor and solve
• Swap variables at the end