f-x-2x-3-x-2-2x-1-f-1-x-

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

f (x) = 2 x^{3} + x^{2} - 2 x - 1

f^{- 1} (x) = \dots

Answered by MJS last updated on 15/Nov/18

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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