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Q.
Let $f ( x )=\frac{ ax + b }{ cx + d } .$ Then $fof ( x )= x$ provided that
Relations and Functions - Part 2
Solution:
Given, $f(x)=\frac{a x+b}{c x+d}$ and fof $(x)=x$
$\Rightarrow f\left(\frac{a x+b}{c x+d}\right)=x $
$\Rightarrow \frac{a\left(\frac{a x+b}{c x+d}\right)+b}{c\left(\frac{a x+b}{c x+d}\right)+d}=x$
$\Rightarrow \frac{ x \left( a ^{2}+ bc \right)+ ab + bd }{ x ( ac + cd )+ bc + d ^{2}}= x$
$\Rightarrow a ^{2}+ bc = bc + d ^{2}, ab + bd =0$ and
$ac + cd =0$
$\Rightarrow d =- a$