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Question Number 213953 by efronzo1 last updated on 22/Nov/24

Answered by mehdee7396 last updated on 22/Nov/24

$$letf\left(x\right)=\frac{ax+b}{cx+d}\Rightarrow f\left(f\left(x\right)\right)=\frac{a\frac{ax+b}{cx+d}+b}{c\frac{ax+b}{cx+d}+d}$$$$=\frac{\frac{a^{2}x+ab}{cx+d}+b}{\frac{acx+bc}{cx+d}+d}=\frac{(a^2+bc)x+ab+bd}{(ac+cd)+bc+d^2}$$$$\Rightarrow a^2+bc=1\mathrm{\&}ac+cd=1\mathrm{\&}ab+bd=1\mathrm{\&}bc+d^2=2$$$$bc==1-a^2\Rightarrow 1-a^2+d^2=2\Rightarrow d^2-a^2=1$$$$abc+bcd=b\mathrm{\&}abc+bcd=c\Rightarrow b=c$$$$\Rightarrow a^2+b^2=1\mathrm{\&}{b}^2+d^2=2$$$$fora=0\mathrm{\&}b=d=c=1$$$$iff\left(x\right)=\frac{1}{x+1}\Rightarrow f\left(f\left(x\right)\right)=\frac{x+1}{x+2}$$$$\Rightarrow f\left(3\right)=\frac{1}{4}$$$$$$

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