f-x-f-x-1-x-1-x-f-x- Tinku Tara June 3, 2023 Relation and Functions 0 Comments FacebookTweetPin Question Number 2205 by prakash jain last updated on 08/Nov/15 f(x)+f(x−1x)=1+xf(x)=? Commented by Yozzi last updated on 08/Nov/15 f(x)+f(1−1x)=1+x⇒f′(x)+1x2f′(1−1x)=1x2f′(x)+f′(x−1x)=x2 Answered by Rasheed Soomro last updated on 10/Nov/15 f(x)+f(x−1x)=1+xf(x)=1+x−f(x−1x)………………………….(1)f(x−1x)=1+x−1x−f(x−1x−1x−1x)[Replacingxbyx−1xin(1)]=1+x−1x−f(−1x−1)………………….(2)f(−1x−1)=1+−1x−1−f(−1x−1−1−1x−1)[Replacingxby−1x−1in(1)]=1−1x−1−f(x)………………………(3)−−−−−−−−−−−−−−−−−−Replacingf(x−1x)byrhsof(2)andf(−1x−1)byrhsof(3)andf(x)byrhsof(1)in(1)f(x)=1+x−f(x−1x)f(x)=1+x−[1+x−1x−f(−1x−1)]f(x)=x−x−1x+f(−1x−1)]=x−x−1x+1−1x−1−f(x)2f(x)=x−x−1x+1−1x−1f(x)=12(x−x−1x−1x−1+1)−−−−−−−−−−−−−−−−−−−−−−−−−VERIFICATION:f(x)=12(x−x−1x−1x−1+1)f(x−1x)=12(x−1x−x−1x−1x−1x−1x−1x−1+1)=12(x−1x+1x−1+x+1)f(x)+f(x−1x)=12(x−x−1x×−1x−1×+1+x−1x×+1x−1×+x+1)=12(2x+2)=x+1 Commented by Rasheed Soomro last updated on 11/Nov/15 f(x)involvesf(x−1x),f(x−1x)involvesf(−1x−1)andf(−1x−1)involvesagainf(x).Thisiscyclic.Atlastwehaveanequationwhichinvolvesonlyf(x)whichgivesdefinitionoff(x).Ithinkthiscyclictypefunctionalequationshowsacategaryoffunctionalequations.Wecanalsotalkoforderofsuchequations. Terms of Service Privacy Policy Contact: info@tinkutara.com FacebookTweetPin Post navigation Previous Previous post: calculus-prove-dx-x-2-pi-2-cosh-x-4-1-Next Next post: Question-67743 Leave a Reply Cancel replyYour email address will not be published. Required fields are marked *Comment * Name * Save my name, email, and website in this browser for the next time I comment.