Find \(f^{-1}(-1)\) for \(f(x)=x^3\) on \(\mathbb{C}\)
📝 Question
Let:
\[ f:\mathbb{C}\to\mathbb{C}, \quad f(x)=x^3 \]
Find \(f^{-1}(-1)\).
✅ Solution
🔹 Step 1: Meaning of \(f^{-1}(-1)\)
Since \(f(x)=x^3\) is not one-one on \(\mathbb{C}\), inverse function does not exist.
Here, \(f^{-1}(-1)\) means the inverse image of \(-1\).
—🔹 Step 2: Solve Equation
\[ f(x)=-1 \]
\[ x^3=-1 \] —
🔹 Step 3: Find Cube Roots of \(-1\)
Write \(-1\) in exponential form:
\[ -1 = e^{i\pi} \]
Cube roots are:
\[ x = e^{i(\pi+2k\pi)/3}, \quad k=0,1,2 \]
So, the roots are:
:contentReference[oaicite:0]{index=0} —🎯 Final Answer
\[ \boxed{f^{-1}(-1)=\left\{-1,\;\frac{1+i\sqrt{3}}{2},\;\frac{1-i\sqrt{3}}{2}\right\}} \]
🚀 Exam Shortcut
- Solve \(x^3=-1\)
- Use exponential form \(e^{i\theta}\)
- Apply formula: \(\theta+2k\pi\)/3
- Total 3 roots in \(\mathbb{C}\)