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

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

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

$$\mathrm{put}b=0$$$$f\left(a\right).f\left(0\right)-f\left(a\right)=a$$$$f\left(a\right)(f\left(0\right)-1)=a$$$$a=b=0$$$${\left(f\left(0\right)\right)}^2-f\left(0\right)=0$$$$\Rightarrow f\left(0\right)=1orf\left(0\right)=0$$$$iff\left(0\right)=0\Rightarrow f\left(a\right)=-a$$$$\mathrm{it}\mathrm{does}\mathrm{not}\mathrm{satisfy}\mathrm{original}\mathrm{equation}.$$$$ab+(a+b)\ne a+b$$$$ivf\left(0\right)=1f\left(a\right)-f\left(a\right)=a\mathrm{not}\mathrm{possible}$$$$\mathrm{Please}\mathrm{recheck}\mathrm{question}.$$$$\mathrm{I}\mathrm{will}\mathrm{check}\mathrm{my}\mathrm{workings}\mathrm{as}\mathrm{well}.$$

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