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The inverse of function:

- $f$ is a bijection ,$f_{A\rightarrow B}$
- $f^{-1}$ is a bijection ,$f_{B\rightarrow A}$
- $f^{-1}_{B\rightarrow A}(x) = y$ iff $f_{A\rightarrow B}(y) = x$

Given that $f(x) = x^{3} + 1$

$f^{-1}(x) = y \ $ iff $\ f(y) = x$

$\implies y^{3} + 1 = x$

$\implies y^{3} = x - 1$

$\implies y = \sqrt[3]{x-1}$

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