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Question Number 187027 by yaslm last updated on 12/Feb/23

Answered by Farhadazizi last updated on 12/Feb/23

$$fog\left(x\right)=f\left(g\left(x\right)\right)=\frac{2g\left(x\right)-4}{g\left(x\right)}=\frac{2\sqrt{x-1}-4}{\sqrt{x-1}}$$$$gof\left(x\right)=g\left(f\left(x\right)\right)=\sqrt{f\left(x\right)-1}=\sqrt{\frac{x-4}{x}}$$$$f\left(x\right)=\frac{2x-4}{x}=2-\frac{4}{x}\Rrightarrow f\left(x\right)-2=-\frac{4}{x}$$$$-\frac{1}{f\left(x\right)-2}=\frac{x}{4}\Rrightarrow x=\frac{-4}{f\left(x\right)-2}\Rrightarrow f^{-1}\left(x\right)=\frac{-4}{x-2}$$$$$$$$g\left(x\right)=\sqrt{x-1}\Rightarrow {\left(g\left(x\right)\right)}^2+1=x$$$$\Rightarrow g^{-1}\left(x\right)=x^2+1$$

Commented by yaslm last updated on 12/Feb/23

thanks sir,

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