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Question
Let C denote the set of all complex numbers. A function f : C → C is defined by f(x) = x3. Write f−1(1).
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Solution
Let f-1 (1) x ......... (1)
⇒ f (x) = 1
⇒ x3 = 1
⇒ x3 - 1 = 0
⇒ ( x - 1 ) ( x2 +x + 1) = 0 [ Using identity : a3 - b3 = (a - b) (a2 + ab + b2)]
⇒ (x -1) (x -ω) (x + ω2 ) = 0 , where ω = `(1± i sqrt3)/2`
⇒ x = -1, -ω, -ω2 (as x ∈ C)
⇒f-1 (-1) = {-1, -ω, -ω2} [ form (1) ]
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