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Question Number 219404 by golsendro last updated on 24/Apr/25

$$\mathrm{given}\mathrm{g}\left(\mathrm{x}\right)=\frac{\mathrm{x}-2023}{\mathrm{x}-1}$$$$\mathrm{find}\left(\mathrm{gogogogogog}\right)\left(2024\right)$$

Commented by kapoorshah last updated on 25/Apr/25

$$g^{-1}\left(x\right)=\frac{x-2023}{x-1}$$$$\left(gogogogogog\right)\left(2024\right)$$$$=\left(go{g}^{-1}ogo{g}^{-1}ogo{g}^{-1}\right)\left(2024\right)$$$$=2024$$$$$$$$$$

Commented by kapoorshah last updated on 25/Apr/25

$$\left(go{g}^{-1}\right)\left(a\right)=a$$

Answered by y0o0o last updated on 24/Apr/25

Answered by mehdee7396 last updated on 26/Apr/25

$$$$$$f\left(x\right)=\frac{x-a}{x-1}\Rightarrow f\left(f\left(x\right)\right)=\frac{\frac{x-a}{x-1}-a}{\frac{x-a}{x-1}-1}$$$$=\frac{x-a-ax+a}{x-a-x+1}=\frac{(1-a)x}{1-a}=x$$$$\Rightarrow f^n\left(x\right)=f\left(x\right);n=2k$$$$\mathrm{\&}{f}^n\left(x\right)=x;n=2k+1$$$$$$$$$$

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