Let-f-x-x-3-2x-6-f-1-x-

Question Number 124580 by Boucatchou last updated on 04/Dec/20

L e t f (x) = x^{3} + 2 x - 6, f^{- 1} (x) = ?

Answered by MJS_new last updated on 04/Dec/20

x = y^{3} + 2 y - 6

y^{3} + 2 y - (x + 6) = 0

D = \frac{p^{3}}{27} + \frac{q^{2}}{4} = \frac{8}{27} + \frac{{(x + 6)}^{2}}{4} > 0 \forall x \in R \Rightarrow we need

{Cardano}^{'} s solution

f^{- 1} (x) = \sqrt[3]{\frac{x}{2} + 3 + \sqrt{\frac{x^{2}}{2} + 6 x + \frac{494}{27}}} - \sqrt[3]{- \frac{x}{2} - 3 + \sqrt{\frac{x^{2}}{2} + 6 x + \frac{494}{27}}}

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