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

🎥 Video Explanation

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📝 Question

Let \(A=\{x\in\mathbb{R}:x\ge1\}\).

\[ f(x)=2^x(x-1) \]

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

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✅ Solution

🔹 Step 1: Let \(y=f(x)\)

\[ y=2^x(x-1) \] —

🔹 Step 2: Try Substitution

Let \(t=x-1\) ⇒ \(x=t+1\)

\[ y=2^{t+1}\cdot t = 2\cdot 2^t \cdot t \]

\[ \frac{y}{2}=t\cdot 2^t \] —

🔹 Step 3: Recognize Form

This is of the form:

\[ t\cdot 2^t = k \]

This cannot be solved using elementary algebra.

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🔹 Step 4: Use Lambert W Function

Rewrite:

\[ t\cdot e^{t\ln2}=\frac{y}{2} \]

\[ (t\ln2)\,e^{t\ln2}=\frac{y\ln2}{2} \]

\[ t\ln2 = W\!\left(\frac{y\ln2}{2}\right) \]

\[ t=\frac{1}{\ln2}W\!\left(\frac{y\ln2}{2}\right) \]

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🔹 Step 5: Back Substitute

\[ x=t+1 \]

\[ f^{-1}(x)=1+\frac{1}{\ln2}W\!\left(\frac{x\ln2}{2}\right) \] —

🔹 Final Answer

\[ \boxed{f^{-1}(x)=1+\frac{1}{\ln2}W\!\left(\frac{x\ln2}{2}\right)} \]