f-x-x-3-ax-f-1-x-

Question Number 175866 by Linton last updated on 08/Sep/22

f (x) = x^{3} + a x

f^{- 1} (x) = ?

Answered by mr W last updated on 08/Sep/22

y = x^{3} + a x

x^{3} + a x - y = 0

x = \sqrt[3]{\sqrt{\frac{y^{2}}{4} + \frac{a^{3}}{27}} + \frac{y}{2}} - \sqrt[3]{\sqrt{\frac{y^{2}}{4} + \frac{a^{3}}{27}} - \frac{y}{2}}

\Rightarrow f^{- 1} (x) = \sqrt[3]{\sqrt{\frac{x^{2}}{4} + \frac{a^{3}}{27}} + \frac{x}{2}} - \sqrt[3]{\sqrt{\frac{x^{2}}{4} + \frac{a^{3}}{27}} - \frac{x}{2}}

Commented by peter frank last updated on 08/Sep/22

please explain 3^{rd} line

Commented by mr W last updated on 09/Sep/22

{c a r d a n o}^{'} s f o r m u l a

s e e Q 89687

Commented by Tawa11 last updated on 15/Sep/22

Great sir

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