If-f-x-x-3-x-1-then-invrse-function-of-f-x-is-

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

$${If}\:{f}\left({x}\right)={x}^{\mathrm{3}} +{x}+\mathrm{1}\:{then}\:{invrse}\:{function} \\ $$$${of}\:{f}\left({x}\right)\:{is}??? \\ $$

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

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

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

Commented by SLVR last updated on 21/Sep/21