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Question Number 125968 by bramlexs22 last updated on 16/Dec/20
Find f(x) such that f(x)+f((1/(1−x))) = x
Findf(x)suchthatf(x)+f(11−x)=x
Answered by liberty last updated on 16/Dec/20
(1) f(x)+f((1/(1−x))) = x replace x→(1/(1−x)) (2)f((1/(1−x)))+f(((x−1)/x))=(1/(1−x)) replace(1/(1−x))→x (3)f(((x−1)/x))+f(x)=((x−1)/x) ⇔(1)+(3)⇒2f(x)+f(((x−1)/x))+f((1/(1−x)))=((x^2 +x−1)/x) ⇒2f(x)+(1/(1−x)) = ((x^2 +x−1)/x) ⇒2f(x)= ((x^2 +x−1)/x)−(1/(1−x)) ⇒2f(x)=((x^2 +x−1)/x)+(1/(x−1))=((x^3 −x^2 +x^2 −x−x+1+x)/(x(x−1))) f(x)=((x^3 −x+1)/(2x(x−1)))
(1)f(x)+f(11−x)=xreplacex→11−x(2)f(11−x)+f(x−1x)=11−xreplace11−x→x(3)f(x−1x)+f(x)=x−1x⇔(1)+(3)⇒2f(x)+f(x−1x)+f(11−x)=x2+x−1x⇒2f(x)+11−x=x2+x−1x⇒2f(x)=x2+x−1x−11−x⇒2f(x)=x2+x−1x+1x−1=x3−x2+x2−x−x+1+xx(x−1)f(x)=x3−x+12x(x−1)

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