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Question Number 179436 by Shrinava last updated on 29/Oct/22

$$\mathrm{f}\left(\mathrm{x}\right)=\frac{1}{\mathrm{x}}\mathrm{and}\mathrm{g}\left(\mathrm{x}\right)=\frac{\mathrm{x}+2}{2\mathrm{x}+1}$$$$\mathrm{Find}:{\mathrm{g}}^{-1}\left(\mathrm{f}\left(3\right)\right)=?$$

Commented by cherokeesay last updated on 29/Oct/22

$$y=\frac{x+2}{2x+1}\iff 2xy+y=x+2\Rightarrow$$$$x(2y-1)=2-y\iff x=\frac{2-y}{2y-1}$$$$\iff g^{-1}\left(x\right)=\frac{2-x}{2x-1}$$$$g^{-1}\left[f\left(3\right)\right]=\frac{2-\frac{1}{3}}{2\times \frac{1}{3}-1}=-5$$

Answered by ARUNG_Brandon_MBU last updated on 29/Oct/22

$$\mathrm{Let}\mathrm{y}=\frac{x+2}{2x+1}\Rightarrow x=\frac{2-\mathrm{y}}{2\mathrm{y}-1}\Rightarrow g^{-1}\left(x\right)=\frac{2-x}{2x-1}$$$$f\left(3\right)=\frac{1}{3}\Rightarrow g^{-1}\left(f\left(3\right)\right)=g^{-1}\left(\frac{1}{3}\right)=-5$$

Answered by mr W last updated on 29/Oct/22

$$f\left(3\right)=\frac{1}{3}$$$$g\left(x\right)=\frac{x+2}{2x+1}=\frac{1}{3}$$$$\Rightarrow x=-5$$$$g(-5)=\frac{1}{3}=f\left(3\right)$$$$\Rightarrow g^{-1}\left(f\left(3\right)\right)=-5$$

Commented by Shrinava last updated on 29/Oct/22

$$\mathrm{thank}\mathrm{you}\mathrm{dear}\mathrm{professors}$$

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