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f-0-R-g-0-R-f-ux-f-uf-x-g-ux-g-u-f-x-f-x-g-x-




Question Number 3886 by 123456 last updated on 23/Dec/15
f:[0,+∞)→R  g:[0,+∞)→R  f(ux)=f(uf(x))  g(ux)=g(u+f(x))  f(x)+g(x)=?
$${f}:\left[\mathrm{0},+\infty\right)\rightarrow\mathbb{R} \\ $$$${g}:\left[\mathrm{0},+\infty\right)\rightarrow\mathbb{R} \\ $$$${f}\left({ux}\right)={f}\left({uf}\left({x}\right)\right) \\ $$$${g}\left({ux}\right)={g}\left({u}+{f}\left({x}\right)\right) \\ $$$${f}\left({x}\right)+{g}\left({x}\right)=? \\ $$
Commented by Yozzii last updated on 24/Dec/15
Let u=1.  ∴ f(x)+g(x)=f(f(x))+g(1+f(x)).  f(x)=f(f(x))  Let f(x)=x⇒f(f(x))=f(x)=x  and f: [0,∞)→R.  ∴g(x)=g(1+x)  Periodic function with period 1.  g(x)=Asin2πx (A∈R) and g: [0,∞)→R.  ⇒g(x+1)=Asin(2π+2πx)=Asin2πx=g(x)  ∴ f(x)+g(x)=x+Asin2πx for example    g(x)=Asin2mπx+Bcos2nπx (n,m∈Z, A,B∈R)  ∵ g(1+x)=Asin(2mπ+2mπx)+Bcos(2nπ+2πnx)  g(1+x)=Asin2mπx+Bcos2nπx=g(x)    f(x)=0 or f(x)=x.
$${Let}\:{u}=\mathrm{1}. \\ $$$$\therefore\:{f}\left({x}\right)+{g}\left({x}\right)={f}\left({f}\left({x}\right)\right)+{g}\left(\mathrm{1}+{f}\left({x}\right)\right). \\ $$$${f}\left({x}\right)={f}\left({f}\left({x}\right)\right) \\ $$$${Let}\:{f}\left({x}\right)={x}\Rightarrow{f}\left({f}\left({x}\right)\right)={f}\left({x}\right)={x} \\ $$$${and}\:{f}:\:\left[\mathrm{0},\infty\right)\rightarrow\mathbb{R}. \\ $$$$\therefore{g}\left({x}\right)={g}\left(\mathrm{1}+{x}\right) \\ $$$${Periodic}\:{function}\:{with}\:{period}\:\mathrm{1}. \\ $$$${g}\left({x}\right)={Asin}\mathrm{2}\pi{x}\:\left({A}\in\mathbb{R}\right)\:{and}\:{g}:\:\left[\mathrm{0},\infty\right)\rightarrow\mathbb{R}. \\ $$$$\Rightarrow{g}\left({x}+\mathrm{1}\right)={Asin}\left(\mathrm{2}\pi+\mathrm{2}\pi{x}\right)={Asin}\mathrm{2}\pi{x}={g}\left({x}\right) \\ $$$$\therefore\:{f}\left({x}\right)+{g}\left({x}\right)={x}+{Asin}\mathrm{2}\pi{x}\:{for}\:{example} \\ $$$$ \\ $$$${g}\left({x}\right)={Asin}\mathrm{2}{m}\pi{x}+{Bcos}\mathrm{2}{n}\pi{x}\:\left({n},{m}\in\mathbb{Z},\:{A},{B}\in\mathbb{R}\right) \\ $$$$\because\:{g}\left(\mathrm{1}+{x}\right)={Asin}\left(\mathrm{2}{m}\pi+\mathrm{2}{m}\pi{x}\right)+{Bcos}\left(\mathrm{2}{n}\pi+\mathrm{2}\pi{nx}\right) \\ $$$${g}\left(\mathrm{1}+{x}\right)={Asin}\mathrm{2}{m}\pi{x}+{Bcos}\mathrm{2}{n}\pi{x}={g}\left({x}\right) \\ $$$$ \\ $$$${f}\left({x}\right)=\mathrm{0}\:{or}\:{f}\left({x}\right)={x}. \\ $$$$ \\ $$

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