Find \(f^{-1}(1)\) for \(f(x)=x^4\) on \(\mathbb{C}\)
📝 Question
Let:
\[ f:\mathbb{C}\to\mathbb{C}, \quad f(x)=x^4 \]
Find \(f^{-1}(1)\).
✅ Solution
🔹 Step 1: Meaning of \(f^{-1}(1)\)
Since \(f(x)=x^4\) 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
\[ f(x)=1 \]
\[ x^4=1 \] —
🔹 Step 3: Find Fourth Roots of Unity
Write \(1\) in exponential form:
\[ 1 = e^{2k\pi i} \]
Fourth roots are:
\[ x = e^{\frac{2k\pi i}{4}}, \quad k=0,1,2,3 \]
So, the roots are:
:contentReference[oaicite:0]{index=0} —🎯 Final Answer
\[ \boxed{f^{-1}(1)=\{1,\,i,\,-1,\,-i\}} \]
🚀 Exam Shortcut
- Solve \(x^4=1\)
- Use roots of unity formula
- Total 4 roots in \(\mathbb{C}\)
- Even powers ⇒ symmetric roots