If f : C C is defined by f(x) = x^4, write f^ -1 (1). - Vidyadip

Question

If $f : C \rightarrow C$ is defined by $f(x) = x^4,$ write $f^{-1}(1).$

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Answer

Let $f^{-1}(1) = x ......(1)$
$\Rightarrow f(x) = 1$
$\Rightarrow x^4 = 1$
$\Rightarrow x^4 - 1 = 0$
$\Rightarrow (x^2 - 1)(x^2 + 1) = 0 [$Using identity: $a^2 - b^2 = (a - b)(a + b)]$
$\Rightarrow (x - 1)(x + 1)(x - i)(x + i) = 0$ where $\text{i}=\sqrt{-1} [$Using identity$: a^2 - b^2 = (a - b)(a + b)]$
$\Rightarrow\ \text{x}=\pm1,\pm\text{i}$
$\Rightarrow f^{-1}(1) = {-1, 1, i, -i} [$from $(1)]$

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