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Question Number 38367 by Zuarkton last updated on 24/Jun/18

If f(x)=x^3 +1 then f^(−1) (x)=?

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

Commented by rahul 19 last updated on 24/Jun/18

(x−1)^(1/3)

$$\left({x}−\mathrm{1}\right)^{\frac{\mathrm{1}}{\mathrm{3}}} \\ $$

Commented by Zuarkton last updated on 24/Jun/18

Solution?

$${Solution}? \\ $$

Answered by MrW3 last updated on 24/Jun/18

y=f(x)=x^3 +1  ⇒x=^3 (√(y−1))  ⇒f^(−1) (x)=^3 (√(x−1))

$${y}={f}\left({x}\right)={x}^{\mathrm{3}} +\mathrm{1} \\ $$$$\Rightarrow{x}=\:^{\mathrm{3}} \sqrt{{y}−\mathrm{1}} \\ $$$$\Rightarrow{f}^{−\mathrm{1}} \left({x}\right)=\:^{\mathrm{3}} \sqrt{{x}−\mathrm{1}} \\ $$

Answered by Rio Mike last updated on 25/Jun/18

 let y = x^3  + 1      x^3  = y − 1     x = (y − 1)^(1/3)     replacing x by f^(−1) (x) and y by x  ⇒  f^(−1) (x)= (x − 1)^(1/3)

$$\:{let}\:{y}\:=\:{x}^{\mathrm{3}} \:+\:\mathrm{1}\: \\ $$$$\:\:\:{x}^{\mathrm{3}} \:=\:{y}\:−\:\mathrm{1} \\ $$$$\:\:\:{x}\:=\:\left({y}\:−\:\mathrm{1}\right)^{\mathrm{1}/\mathrm{3}} \\ $$$$\:\:{replacing}\:{x}\:{by}\:{f}^{−\mathrm{1}} \left({x}\right)\:{and}\:{y}\:{by}\:{x} \\ $$$$\Rightarrow\:\:{f}^{−\mathrm{1}} \left({x}\right)=\:\left({x}\:−\:\mathrm{1}\right)^{\mathrm{1}/\mathrm{3}} \\ $$

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