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Question
If f : C → C is defined by f(x) = x2, write f−1 (−4). Here, C denotes the set of all complex numbers.
Solution
Let f−1 (−4) = x ... (1)
⇒ f(x) = −4
⇒ x2 = −4
⇒ x2 + 4 = 0
⇒ ( x+2i ) ( x−2i ) = 0 [using the identity: a2+b2=( a − ib ) (a + ib)]
⇒ x = ±2i [as x∈C]
⇒ f−1 (25) = {−2i, 2i} [from (1)]
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