Find \(f^{-1}(x)\)
🎥 Video Explanation
📝 Question
Let \(A=\{x\in\mathbb{R}:x\le1\}\)
\[ f(x)=x(2-x) \]
- (a) \(1+\sqrt{1-x}\)
- (b) \(1-\sqrt{1-x}\)
- (c) \(\sqrt{1-x}\)
- (d) \(1\pm\sqrt{1-x}\)
✅ Solution
🔹 Step 1: Rewrite Function
\[ f(x)=2x-x^2 \]
\[ =1-(x-1)^2 \]
—🔹 Step 2: Let \(y=f(x)\)
\[ y=1-(x-1)^2 \] —
🔹 Step 3: Solve for \(x\)
\[ (x-1)^2=1-y \]
\[ x-1=\pm\sqrt{1-y} \]
\[ x=1\pm\sqrt{1-y} \] —
🔹 Step 4: Apply Domain Restriction
Given \(x \le 1\), so:
\[ x=1-\sqrt{1-y} \]
—🔹 Step 5: Write Inverse
\[ f^{-1}(x)=1-\sqrt{1-x} \] —
🔹 Final Answer
\[ \boxed{\text{Option (b)}} \]