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Question
Let f be a function from C (set of all complex numbers) to itself given by f(x) = x3. Write f−1 (−1).
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Solution
\[Let f^{- 1} \left( - 1 \right) = x . . . \left( 1 \right)\]
\[ \Rightarrow f\left( x \right) = - 1\]
\[ \Rightarrow x^3 = - 1\]
\[ \Rightarrow x^3 + 1 = 0\]
\[ \Rightarrow \left( x + 1 \right)\left( x^2 - x + 1 \right) = 0 \left[ \text{using the identity}: a^3 + b^3 = \left( a + b \right)\left( a^2 - ab + b^2 \right) \right]\]
\[ \Rightarrow \left( x + 1 \right)\left( x + \omega \right)\left( x + \omega^2 \right) = 0, where \omega = \frac{1 \pm i\sqrt{3}}{2} \]
\[ \Rightarrow x = - 1, - \omega, - \omega^2 \left( asx \in C \right)\]
\[ \Rightarrow f^{- 1} \left( - 1 \right) = \left\{ - 1, - \omega, - \omega^2 \right\} [from\left( 1 \right)]\]
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