Question and Answers Forum

All Questions      Topic List

Integration Questions

Previous in All Question      Next in All Question      

Previous in Integration      Next in Integration      

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

f(x)=x^3 +ax  f^(−1) (x)=?

$${f}\left({x}\right)={x}^{\mathrm{3}} +{ax} \\ $$$${f}^{−\mathrm{1}} \left({x}\right)=? \\ $$

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

y=x^3 +ax  x^3 +ax−y=0  x=(((√((y^2 /4)+(a^3 /(27))))+(y/2)))^(1/3) −(((√((y^2 /4)+(a^3 /(27))))−(y/2)))^(1/3)   ⇒f^(−1) (x)=(((√((x^2 /4)+(a^3 /(27))))+(x/2)))^(1/3) −(((√((x^2 /4)+(a^3 /(27))))−(x/2)))^(1/3)

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

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

please explain 3^(rd ) line

$$\mathrm{please}\:\mathrm{explain}\:\mathrm{3}^{\mathrm{rd}\:} \mathrm{line} \\ $$

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

cardano′s formula  see Q89687

$${cardano}'{s}\:{formula} \\ $$$${see}\:{Q}\mathrm{89687} \\ $$

Commented by Tawa11 last updated on 15/Sep/22

Great sir

$$\mathrm{Great}\:\mathrm{sir} \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com