If f : C → C is defined by f(x) = x4, write f-1(1). - Vidyadip

Question

If f : C → C is defined by f(x) = x⁴, write f^-1(1).

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Answer

Let f^-1(1) = x ......(1)
⇒ f(x) = 1
⇒ x⁴ = 1
⇒ x⁴ - 1 = 0
⇒ (x² - 1)(x² + 1) = 0 [Using identity: a² - b² = (a - b)(a + b)]
⇒ (x - 1)(x + 1)(x - i)(x + i) = 0 where $\text{i}=\sqrt{-1}$ [Using identity: a² - b² = (a - b)(a + b)]
$\Rightarrow\ \text{x}=\pm1,\pm\text{i}$
⇒ f^-1(1) = {-1, 1, i, -i} [from (1)]

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