Question Number 3654 by prakash jain last updated on 17/Dec/15
$${f}\left({xf}\left({y}\right)+{x}\right)={xy}+{f}\left({x}\right) \\ $$$${f}:\mathbb{R}\rightarrow\mathbb{R} \\ $$$${f}\left({x}\right)=? \\ $$
Commented by RasheedSindhi last updated on 18/Dec/15
$${f}\left({xf}\left({y}\right)+{x}\right)={xy}+{f}\left({x}\right)….\left({i}\right) \\ $$$${Let}\:{y}=\mathrm{0} \\ $$$${f}\left({xf}\left(\mathrm{0}\right)+{x}\right)={x}\left(\mathrm{0}\right)+{f}\left({x}\right) \\ $$$${f}\left({x}\right)={f}\left(\:{xf}\left(\mathrm{0}\right)+{f}\left({x}\right)\:\right)……..\left({ii}\right) \\ $$$${f}\left(\mathrm{0}\right)={f}\left(\:\mathrm{0}.{f}\left(\mathrm{0}\right)+{f}\left(\mathrm{0}\right)\right) \\ $$$$\:\:\:\:\:\:\:\:\:{f}\left(\mathrm{0}\right)={f}\left(\:{f}\left(\mathrm{0}\right)\:\right) \\ $$$${x}\leftrightharpoons{y}\:{in}\:\left({i}\right) \\ $$$${f}\left({yf}\left({x}\right)+{y}\right)={xy}+{f}\left({y}\right)…….\left({iii}\right) \\ $$$${y}={x}\:{in}\:\left({i}\right) \\ $$$${f}\left(\:{xf}\left({x}\right)+{x}\:\right)={x}^{\mathrm{2}} +{f}\left({x}\right) \\ $$$${f}\left({x}\right)={f}\left(\:{xf}\left({x}\right)+{x}\:\right)−{x}^{\mathrm{2}} \\ $$$${f}\left(\mathrm{1}\right)={f}\left(\:{f}\left(\mathrm{1}\right)+\mathrm{1}\:\right)−\mathrm{1} \\ $$$${f}\left(\mathrm{1}\right)+\mathrm{1}={f}\left(\:{f}\left(\mathrm{1}\right)+\mathrm{1}\:\right) \\ $$
Commented by prakash jain last updated on 18/Dec/15
$$\mathrm{based}\:\mathrm{on}\:\mathrm{above} \\ $$$${f}\left({x}\right)={x}\:\mathrm{is}\:\mathrm{one}\:\mathrm{solution},\:\mathrm{are}\:\mathrm{there}\:\mathrm{any}\:\mathrm{other} \\ $$$$\mathrm{solutions}? \\ $$
Commented by prakash jain last updated on 18/Dec/15
$$\mathrm{Let}\:\mathrm{us}\:\mathrm{say}\:{f}\left({x}\right)\neq{x}\:\mathrm{and}\:{f}\left({y}\right)=\mathrm{0}\:{for}\:{some}\:{y}. \\ $$$${f}\left({x}\right)={xf}^{−\mathrm{1}} \left({y}\right)+{f}\left({x}\right) \\ $$$${f}^{−\mathrm{1}} \left({y}\right)=\mathrm{0} \\ $$$${so}\:{if}\:{f}\left({x}\right)\neq{x}\:{then}\:{either} \\ $$$${f}\left({x}\right)\neq\mathrm{0}\:\mathrm{for}\:\mathrm{all}\:{x}\in\mathbb{R}\:\mathrm{or}\:{f}\left(\mathrm{0}\right)=\mathrm{0}. \\ $$