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Question Number 1278 by Rasheed Soomro last updated on 19/Jul/15
Determine f(x) when f( f(x))=x .
Determinef(x)whenf(f(x))=x.
Commented by prakash jain last updated on 19/Jul/15
f(x)=((ax+b)/(cx+d)) f(f(x))=((a((ax+b)/(cx+d))+b)/(c((ax+b)/(cx+d))+d))=x a^2 x+ab+bcx+bd=c(a+d)x^2 +(bc+d^2 )x Comparing coefficients a+d=0 f(x)=((ax+b)/(cx−a)) f(f(x))=((a((ax+b)/(cx−a))+b)/(c((ax+b)/(cx−a))−a))=((a^2 x+ab+bcx−ab)/(acx+bc−acx+a^2 ))=x More solution can be found.
f(x)=ax+bcx+df(f(x))=aax+bcx+d+bcax+bcx+d+d=xa2x+ab+bcx+bd=c(a+d)x2+(bc+d2)xComparingcoefficientsa+d=0f(x)=ax+bcx−af(f(x))=aax+bcx−a+bcax+bcx−a−a=a2x+ab+bcx−abacx+bc−acx+a2=xMoresolutioncanbefound.
Commented by prakash jain last updated on 19/Jul/15
f(x)=x f(x)=k−x f(x)=(k/x) Also more solutions are possible
f(x)=xf(x)=k−xf(x)=kxAlsomoresolutionsarepossible
Answered by prakash jain last updated on 19/Jul/15
Some solutions are given in comments but more solutions exist.
Somesolutionsaregivenincommentsbutmoresolutionsexist.

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