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Question
If f : C → C is defined by f(x) = x4, write f−1 (1).
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Solution
\[Let f^{- 1} \left( 1 \right) = x . . . \left( 1 \right)\]
\[ \Rightarrow f\left( x \right) = 1\]
\[ \Rightarrow x^4 = 1\]
\[ \Rightarrow x^4 - 1 = 0\]
\[ \Rightarrow \left( x^2 - 1 \right)\left( x^2 + 1 \right) = 0 \left[ \text{using identity}: a^2 - b^2 = \left( a - b \right)\left( a + b \right) \right]\]
\[ \Rightarrow \left( x - 1 \right)\left( x + 1 \right)\left( x - i \right)\left( x + i \right) = 0, \text{where} i = \sqrt{- 1} \left[ \text{using identity}: a^2 - b^2 = \left( a - b \right)\left( a + b \right) \right]\]
\[ \Rightarrow x = \pm 1, \pm i \]
\[ \Rightarrow f^{- 1} \left( 1 \right) = \left\{ - 1, 1, i, - i \right\} [\text{from}\left( 1 \right)]\]
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