Find \(f^{-1}(-1)\) for \(f(x)=(x-2)^3\) on \(\mathbb{C}\)
📝 Question
Let:
\[ f:\mathbb{C}\to\mathbb{C}, \quad f(x)=(x-2)^3 \]
Find \(f^{-1}(-1)\).
✅ Solution
🔹 Step 1: Meaning of \(f^{-1}(-1)\)
Here, \(f^{-1}(-1)\) means the inverse image of \(-1\).
—🔹 Step 2: Solve Equation
\[ f(x)=-1 \]
\[ (x-2)^3=-1 \] —
🔹 Step 3: Find Cube Roots
Let \(y=x-2\). Then:
\[ y^3=-1 \]
The cube roots of \(-1\) are:
:contentReference[oaicite:0]{index=0} —🔹 Step 4: Find \(x\)
\[ x=y+2 \]
So,
\[ x=1,\quad 2+\frac{1+i\sqrt{3}}{2},\quad 2+\frac{1-i\sqrt{3}}{2} \] —
🎯 Final Answer
\[ \boxed{ f^{-1}(-1)=\left\{ 1,\; \frac{5+i\sqrt{3}}{2},\; \frac{5-i\sqrt{3}}{2} \right\} } \]
🚀 Exam Shortcut
- Substitute \(y=x-2\)
- Solve \(y^3=-1\)
- Add 2 to all roots
- Final answer = shifted roots