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Question Number 145641 by physicstutes last updated on 06/Jul/21

$$\mathrm{Let}\mathrm{g}:\mathbb{R}\to \mathbb{R}\mathrm{be}\mathrm{given}\mathrm{by}\mathrm{g}\left(x\right)=3+4x.\mathrm{Prove}\mathrm{by}\mathrm{induction}$$$$\mathrm{that},\mathrm{for}\mathrm{all}\mathrm{positive}\mathrm{integers}n,$$$${\mathrm{g}}^n\left(x\right)=(4^n-1)+4^n\left(x\right).$$$$\mathrm{If}\mathrm{for}\mathrm{every}\mathrm{positive}\mathrm{integer}k,\mathrm{we}\mathrm{inteprete}{\mathrm{g}}^{-k}\mathrm{as}\mathrm{the}\mathrm{inverse}$$$$\mathrm{of}\mathrm{the}\mathrm{function}{\mathrm{g}}^k.\mathrm{Prove}\mathrm{that}\mathrm{the}\mathrm{above}\mathrm{formula}\mathrm{holds}\mathrm{alsl}$$$$\mathrm{for}\mathrm{all}\mathrm{negative}\mathrm{integers}n.$$

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