Question Number 2272 by Rasheed Soomro last updated on 12/Nov/15
Answered by Rasheed Soomro last updated on 16/Nov/15
![f(((−1)/(x−1)))+f(((x−1)/x))=((2x−1)/x) In order to get equation involving f(x), we have to replace ((−1)/(x−1)) (or ((x−1)/x) ) by x : Let ((−1)/(x−1))=y⇒x=((y−1)/y) f(((−1)/(x−1)))+f(((x−1)/x))=((2x−1)/x) ⇒f(y)+f(((((y−1)/y)−1)/((y−1)/y)))=((2(((y−1)/y))−1)/((y−1)/y)) ⇒f(y)+f(((−1)/(y−1))) =((y−2)/(y−1)) ⇒f(x)+f(((−1)/(x−1))) =((x−2)/(x−1)) [replacing y by x ] ⇒f(x)=((x−2)/(x−1))−f(((−1)/(x−1)))........................(1) f(((−1)/(x−1)))=((((−1)/(x−1))−2)/(((−1)/(x−1))−1))−f(((−1)/(((−1)/(x−1))−1))) [ replace x by ((−1)/(x−1)) in (1)] =((2x−1)/x)−f(((x−1)/x)) [ replace x by ((−1)/(x−1)) in (1)] f(((x−1)/x))=((((x−1)/x)−2)/(((x−1)/x)−1))−f(((−1)/(((x−1)/x)−1))) [replace x by ((x−1)/x) in (1) ] =x+1−f(x) [replace x by ((x−1)/x) in (1) ] −−−−−−−−−−−−−−−−−−−−− From (1) f(x)=((x−2)/(x−1))−f(((−1)/(x−1))) =((x−2)/(x−1))−f(((−1)/(x−1))) =((x−2)/(x−1))−(((2x−1)/x)−f(((x−1)/x))) =((x−2)/(x−1))−((2x−1)/x)+x+1−f(x) 2f(x)=((x−2)/(x−1))−((2x−1)/x)+x+1 f(x)=(1/2)(((x−2)/(x−1))−((2x−1)/x)+x+1)](Q2328.png)