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Question Number 165376 by MikeH last updated on 31/Jan/22

$$\mathrm{Verify}\mathrm{wether}f\mathrm{is}\mathrm{invertible}$$$$f\left(x\right)={(1+2x)}^3$$

Answered by Rasheed.Sindhi last updated on 31/Jan/22

$$f\left(x\right)={(1+2x)}^3$$$$y={(1+2x)}^3$$$$1+2x=\sqrt[3]{y}$$$$x=\frac{\sqrt[3]{y}-1}{2}$$$$f^{-1}\left(x\right)=\frac{\sqrt[3]{x}-1}{2}$$

Commented by aleks041103 last updated on 01/Feb/22

$$Ifx\in \mathbb{R},g\left(x\right)=x^{3}isbijective.$$$$Forallx,l\left(x\right)=1+2xisbijective.$$$$Thereforef=g\circ lisacomposition$$$$of2bijectivefunctions\Rightarrow fisbijective.$$$$\Rightarrow fisinvertible,butonlyon\mathbb{R}.$$

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