Question

If f : C → C is defined by f(x) = (x − 2)³, write f⁻¹ (−1).

Answer 1

\[Let f^{- 1} \left( - 1 \right) = x . . . \left( 1 \right)\]
\[ \Rightarrow f\left( x \right) = - 1\]
\[ \Rightarrow \left( x - 2 \right)^3 = - 1\]
\[ \Rightarrow x - 2 = - 1 \text{or} - \omega or - \omega^2 \left( \text{as the roots of } \left( - 1 \right)^\frac{1}{3} are - 1, - \omega \text{ and } - \omega^2 , \text{where } \omega = \frac{1 \pm i\sqrt{3}}{2} \right)\]
\[ \Rightarrow x = - 1 + 2 or 2 - \omega or 2 - \omega^2 = 1, 2 - \omega, 2 - \omega\]
\[ \Rightarrow f^{- 1} \left( - 1 \right) = \left\{ 1, 2 - \omega, 2 - \omega^2 \right\} [\text{from}\left( 1 \right)]\]
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