Question and Answers Forum

All Questions      Topic List

Relation and Functions Questions

Previous in All Question      Next in All Question      

Previous in Relation and Functions      Next in Relation and Functions      

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

          Let  f(x)=x^3 +2x−6,    f^(−1) (x)=?

$$ \\ $$$$\:\:\:\:\:\:\:\:{Let}\:\:{f}\left({x}\right)={x}^{\mathrm{3}} +\mathrm{2}{x}−\mathrm{6},\:\:\:\:{f}^{−\mathrm{1}} \left({x}\right)=? \\ $$

Answered by MJS_new last updated on 04/Dec/20

x=y^3 +2y−6  y^3 +2y−(x+6)=0  D=(p^3 /(27))+(q^2 /4)=(8/(27))+(((x+6)^2 )/4)>0 ∀x∈R ⇒ we need  Cardano′s solution  f^(−1) (x)=(((x/2)+3+(√((x^2 /2)+6x+((494)/(27))))))^(1/3) −((−(x/2)−3+(√((x^2 /2)+6x+((494)/(27))))))^(1/3)

$${x}={y}^{\mathrm{3}} +\mathrm{2}{y}−\mathrm{6} \\ $$$${y}^{\mathrm{3}} +\mathrm{2}{y}−\left({x}+\mathrm{6}\right)=\mathrm{0} \\ $$$${D}=\frac{{p}^{\mathrm{3}} }{\mathrm{27}}+\frac{{q}^{\mathrm{2}} }{\mathrm{4}}=\frac{\mathrm{8}}{\mathrm{27}}+\frac{\left({x}+\mathrm{6}\right)^{\mathrm{2}} }{\mathrm{4}}>\mathrm{0}\:\forall{x}\in\mathbb{R}\:\Rightarrow\:\mathrm{we}\:\mathrm{need} \\ $$$$\mathrm{Cardano}'\mathrm{s}\:\mathrm{solution} \\ $$$${f}^{−\mathrm{1}} \left({x}\right)=\sqrt[{\mathrm{3}}]{\frac{{x}}{\mathrm{2}}+\mathrm{3}+\sqrt{\frac{{x}^{\mathrm{2}} }{\mathrm{2}}+\mathrm{6}{x}+\frac{\mathrm{494}}{\mathrm{27}}}}−\sqrt[{\mathrm{3}}]{−\frac{{x}}{\mathrm{2}}−\mathrm{3}+\sqrt{\frac{{x}^{\mathrm{2}} }{\mathrm{2}}+\mathrm{6}{x}+\frac{\mathrm{494}}{\mathrm{27}}}} \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com