If f : R → R is given by f(x) = x^3, write f^ -1 (1). - Vidyadip

Question

If $f : R → R$ is given by $f(x) = x^3$, write $f^{-1}(1)$.

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Answer

Let $f^{-1}(1) = x$ .....(1)
$\Rightarrow f(x) = 1$
$\Rightarrow x^3 = 1$
$\Rightarrow x^3 - 1 = 0$
$\Rightarrow (x - 1)(x^2 + x + 1) = 0$ [Using the identity: $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$]
$\Rightarrow x = 1 (\text{as x}\in\text{R})$
$\Rightarrow f^{-1}(1) = {1}$ [from (1)]

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