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Question Number 56921 by rahul 19 last updated on 26/Mar/19
$${Given}\:: \\ $$$${f}\left({xy}\right)={f}\left({x}\right).{f}\left({y}\right)\forall{x},{y}\epsilon\mathbb{R}\:{and}\:{f}\left(\mathrm{0}\right)\neq\mathrm{0} \\ $$$${then}\:{f}\left({x}\right)=? \\ $$
Answered by mr W last updated on 26/Mar/19
$${f}\left({xy}\right)={f}\left({x}\right){f}\left({y}\right) \\ $$$${with}\:{y}=\mathrm{0} \\ $$$${f}\left(\mathrm{0}\right)={f}\left({x}\right){f}\left(\mathrm{0}\right) \\ $$$${since}\:{f}\left(\mathrm{0}\right)\neq\mathrm{0}\:\Rightarrow{f}\left({x}\right)=\mathrm{1} \\ $$
Commented by rahul 19 last updated on 26/Mar/19
$${Sir},\:{how}\:{to}\:{find}\:{value}\:{of}\:{n}\:{if}\:{we}\:{assumed} \\ $$$${f}\left({x}\right)={x}^{{n}} \:? \\ $$
Commented by mr W last updated on 27/Mar/19
$${f}\left({x}\right)\neq{x}^{{n}} ,\:{since}\:{f}\left(\mathrm{0}\right)\neq\mathrm{0}. \\ $$$${f}\left({x}\right)=\mathrm{1}\:{is}\:{the}\:{only}\:{one}\:{solution}. \\ $$
Commented by rahul 19 last updated on 27/Mar/19
thank you sir!