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

🎥 Video Explanation

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📝 Question

Let \( f:[1,\infty) \to [2,\infty) \),

\[ f(x)=x+\frac{1}{x} \]

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

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✅ Solution

🔹 Step 1: Let \(y=f(x)\)

\[ y=x+\frac{1}{x} \] —

🔹 Step 2: Multiply

\[ yx=x^2+1 \]

\[ x^2-yx+1=0 \] —

🔹 Step 3: Solve Quadratic

\[ x=\frac{y\pm\sqrt{y^2-4}}{2} \] —

🔹 Step 4: Choose Correct Sign

Given domain \(x \ge 1\), take positive root:

\[ x=\frac{y+\sqrt{y^2-4}}{2} \] —

🔹 Step 5: Replace \(y\) by \(x\)

\[ f^{-1}(x)=\frac{x+\sqrt{x^2-4}}{2} \] —

🔹 Final Answer

\[ \boxed{f^{-1}(x)=\frac{x+\sqrt{x^2-4}}{2}} \]