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Question Number 93464 by i jagooll last updated on 13/May/20

If f a function such that f(a).f(b)−f(a+b)=a+b. find the value of f(2019)

$$\mathrm{If}\:\mathrm{f}\:\mathrm{a}\:\mathrm{function}\:\mathrm{such}\:\mathrm{that}\: \\ $$$$\mathrm{f}\left(\mathrm{a}\right).\mathrm{f}\left(\mathrm{b}\right)−\mathrm{f}\left(\mathrm{a}+\mathrm{b}\right)=\mathrm{a}+\mathrm{b}. \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\mathrm{f}\left(\mathrm{2019}\right)\: \\ $$

Answered by prakash jain last updated on 13/May/20

put b=0 f(a).f(0)−f(a)=a f(a)(f(0)−1)=a a=b=0 (f(0))^2 −f(0)=0 ⇒f(0)=1 or f(0)=0 if f(0)=0⇒f(a)=−a it does not satisfy original equation. ab+(a+b)≠a+b iv f(0)=1 f(a)−f(a)=a not possible Please recheck question. I will check my workings as well.

$$\mathrm{put}\:{b}=\mathrm{0} \\ $$$${f}\left({a}\right).{f}\left(\mathrm{0}\right)−{f}\left({a}\right)={a} \\ $$$${f}\left({a}\right)\left({f}\left(\mathrm{0}\right)−\mathrm{1}\right)={a} \\ $$$${a}={b}=\mathrm{0} \\ $$$$\left({f}\left(\mathrm{0}\right)\right)^{\mathrm{2}} −{f}\left(\mathrm{0}\right)=\mathrm{0} \\ $$$$\Rightarrow{f}\left(\mathrm{0}\right)=\mathrm{1}\:{or}\:{f}\left(\mathrm{0}\right)=\mathrm{0} \\ $$$${if}\:{f}\left(\mathrm{0}\right)=\mathrm{0}\Rightarrow{f}\left({a}\right)=−{a} \\ $$$$\:\:\:\:\:\mathrm{it}\:\mathrm{does}\:\mathrm{not}\:\mathrm{satisfy}\:\mathrm{original}\:\mathrm{equation}. \\ $$$$\:\:\:\:{ab}+\left({a}+{b}\right)\neq{a}+{b} \\ $$$${iv}\:{f}\left(\mathrm{0}\right)=\mathrm{1}\:{f}\left({a}\right)−{f}\left({a}\right)={a}\:\mathrm{not}\:\mathrm{possible} \\ $$$$\mathrm{Please}\:\mathrm{recheck}\:\mathrm{question}. \\ $$$$\mathrm{I}\:\mathrm{will}\:\mathrm{check}\:\mathrm{my}\:\mathrm{workings}\:\mathrm{as}\:\mathrm{well}. \\ $$

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