f(x)=2x^3 +x^2 −2x−1 f^(−1) (x)=...


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Question Number 47778 by gunawan last updated on 14/Nov/18

$f (x) = 2 x^{3} + x^{2} - 2 x - 1$ $f^{- 1} (x) = . . .$
	Answered by MJS last updated on 15/Nov/18
	$2y^3 +y^2 −2y−x−1=0 y^3 +(1/2)y^2 −y−((x+1)/2)=0 y=z−(1/6) z^3 −((13)/(12))z−((54x+35)/(108))=0 now it depends on the value of x p=−((13)/(12)); q=−((54x+35)/(108)) D=(p^3 /(27))+(q^2 /4)=(1/(16))x^2 +((35)/(432))x−(1/(48)) case 1 D<0 ⇒ 3 real solutions ⇒ trigonometric method −((35)/(54))−((13(√(13)))/(54))<x<−((35)/(54))+((13(√(13)))/(54)) z=2(√(−(p/3)))sin (((2πk)/3)+(1/3)arcsin (((9q)/(2p^2 ))(√(−(p/3))))) with k=0, 1, 2 z= { ((−((√(13))/3)sin ((1/3)arcsin (((54x+35)(√(13)))/(169))))),((((√(13))/3)sin ((π/3)+(1/3)arcsin (((54x+35)(√(13)))/(169))))),((−((√(13))/3)cos ((π/6)+(1/3)arcsin (((54x+35)(√(13)))/(169))))) :} f^(−1) : { ((y=−(1/6)−((√(13))/3)sin ((1/3)arcsin (((54x+35)(√(13)))/(169))))),((y=−(1/6)+((√(13))/3)sin ((π/3)+(1/3)arcsin (((54x+35)(√(13)))/(169))))),((y=−(1/6)−((√(13))/3)cos ((π/6)+(1/3)arcsin (((54x+35)(√(13)))/(169))))) :} case 2 D=0 ⇒ 2 real solutions ⇒ Cardano′s method x=−((35)/(54))±((13(√(13)))/(54)) z=((−(q/2)+(√D)))^(1/3) +((−(q/2)−(√D)))^(1/3) =2((−(q/2)))^(1/3) z=±((√(13))/3) f^(−1) (x): y=−(1/6)±((√(13))/3) case 3 D>0 ⇒ 1 real solution ⇒ Cardano′s method x<−((35)/(54))−((13(√(13)))/(54)) ∨ x>−((35)/(54))+((13(√(13)))/(54)) z=((−(q/2)+(√D)))^(1/3) +((−(q/2)−(√D)))^(1/3) z=(1/6)((54x+35+6(√(3(27x^2 +35x−9)))))^(1/3) +(1/6)((54x+35−6(√(3(27x^2 +35x−9)))))^(1/3) f^(−1) (x): (1/6)(−1+((54x+35+6(√(3(27x^2 +35x−9)))))^(1/3) +((54x+35−6(√(3(27x^2 +35x−9)))))^(1/3) )$
	$2 y^{3} + y^{2} - 2 y - x - 1 = 0$ $y^{3} + \frac{1}{2} y^{2} - y - \frac{x + 1}{2} = 0$ $y = z - \frac{1}{6}$ $z^{3} - \frac{13}{12} z - \frac{54 x + 35}{108} = 0$ $now it depends on the value of x$ $p = - \frac{13}{12}; q = - \frac{54 x + 35}{108}$ $D = \frac{p^{3}}{27} + \frac{q^{2}}{4} = \frac{1}{16} x^{2} + \frac{35}{432} x - \frac{1}{48}$ $case 1$ $D < 0 \Rightarrow 3 real solutions \Rightarrow trigonometric method$ $- \frac{35}{54} - \frac{13 \sqrt{13}}{54} < x < - \frac{35}{54} + \frac{13 \sqrt{13}}{54}$ $z = 2 \sqrt{- \frac{p}{3}} \sin (\frac{2 π k}{3} + \frac{1}{3} \arcsin (\frac{9 q}{2 p^{2}} \sqrt{- \frac{p}{3}})) with k = 0, 1, 2$ $z = {\begin{cases} - \frac{\sqrt{13}}{3} \sin (\frac{1}{3} \arcsin \frac{(54 x + 35) \sqrt{13}}{169}) \\ \frac{\sqrt{13}}{3} \sin (\frac{π}{3} + \frac{1}{3} \arcsin \frac{(54 x + 35) \sqrt{13}}{169}) \\ - \frac{\sqrt{13}}{3} \cos (\frac{π}{6} + \frac{1}{3} \arcsin \frac{(54 x + 35) \sqrt{13}}{169}) \end{cases}$ $f^{- 1} : {\begin{cases} y = - \frac{1}{6} - \frac{\sqrt{13}}{3} \sin (\frac{1}{3} \arcsin \frac{(54 x + 35) \sqrt{13}}{169}) \\ y = - \frac{1}{6} + \frac{\sqrt{13}}{3} \sin (\frac{π}{3} + \frac{1}{3} \arcsin \frac{(54 x + 35) \sqrt{13}}{169}) \\ y = - \frac{1}{6} - \frac{\sqrt{13}}{3} \cos (\frac{π}{6} + \frac{1}{3} \arcsin \frac{(54 x + 35) \sqrt{13}}{169}) \end{cases}$ $case 2$ $D = 0 \Rightarrow 2 real solutions \Rightarrow {Cardano}^{'} s method$ $x = - \frac{35}{54} \pm \frac{13 \sqrt{13}}{54}$ $z = \sqrt[3]{- \frac{q}{2} + \sqrt{D}} + \sqrt[3]{- \frac{q}{2} - \sqrt{D}} = 2 \sqrt[3]{- \frac{q}{2}}$ $z = \pm \frac{\sqrt{13}}{3}$ $f^{- 1} (x) : y = - \frac{1}{6} \pm \frac{\sqrt{13}}{3}$ $case 3$ $D > 0 \Rightarrow 1 real solution \Rightarrow {Cardano}^{'} s method$ $x < - \frac{35}{54} - \frac{13 \sqrt{13}}{54} \lor x > - \frac{35}{54} + \frac{13 \sqrt{13}}{54}$ $z = \sqrt[3]{- \frac{q}{2} + \sqrt{D}} + \sqrt[3]{- \frac{q}{2} - \sqrt{D}}$ $z = \frac{1}{6} \sqrt[3]{54 x + 35 + 6 \sqrt{3 (27 x^{2} + 35 x - 9)}} + \frac{1}{6} \sqrt[3]{54 x + 35 - 6 \sqrt{3 (27 x^{2} + 35 x - 9)}}$ $f^{- 1} (x) : \frac{1}{6} (- 1 + \sqrt[3]{54 x + 35 + 6 \sqrt{3 (27 x^{2} + 35 x - 9)}} + \sqrt[3]{54 x + 35 - 6 \sqrt{3 (27 x^{2} + 35 x - 9)}})$
	Commented by gunawan last updated on 21/Nov/18

	$thanks$
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