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Question Number 154725 by SLVR last updated on 21/Sep/21

$$Iff\left(x\right)=x^3+x+1theninvrsefunction$$$$off\left(x\right)is???$$

Answered by mr W last updated on 21/Sep/21

$$y=x^3+x+1$$$$x^3+x+1-y=0$$$$x=\sqrt[3]{\sqrt{{\left(\frac{1}{3}\right)}^3+{\left(\frac{1-y}{2}\right)}^2}-\left(\frac{1-y}{2}\right)}-\sqrt[3]{\sqrt{{\left(\frac{1}{3}\right)}^3+{\left(\frac{1-y}{2}\right)}^2}+\left(\frac{1-y}{2}\right)}$$$$x=\sqrt[3]{\sqrt{\frac{1}{27}+\frac{{(y-1)}^2}{4}}+\frac{y-1}{2}}-\sqrt[3]{\sqrt{\frac{1}{27}+\frac{{(y-1)}^2}{4}}-\frac{y-1}{2}}$$$$\Rightarrow f^{-1}\left(x\right)=\sqrt[3]{\sqrt{\frac{1}{27}+\frac{{(x-1)}^2}{4}}+\frac{x-1}{2}}-\sqrt[3]{\sqrt{\frac{1}{27}+\frac{{(x-1)}^2}{4}}-\frac{x-1}{2}}$$

Commented by SLVR last updated on 21/Sep/21

$$Great...dearprofessor...mr.W$$$$Ineedtoknowhowx=in3rdstep?$$$$kindly...explain$$

Commented by mr W last updated on 21/Sep/21

$${Cardano}^{\prime }sformulaforcubicequation.$$$$seeQ89687$$

Commented by SLVR last updated on 21/Sep/21

$$Thankyou...sir...so..kindofyou$$

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