Give some examples of functions, $F$ and $G$ such that $$x=\sqrt{F(x)+G(x)\sqrt{F(x+n)}}-\sqrt{F(x+n)}.$$ $n$ can be a constant.

[Edit]: with $n\gt{0}$

  • $\begingroup$ For $x\ge 0$, $F(x)=x^2, \,\,G(x)=3|x|,\,\, n=0$ $\endgroup$ – Svetoslav Jul 29 '15 at 14:00
  • $\begingroup$ Thanks but I just edited my question. $\endgroup$ – tyobrien Jul 29 '15 at 14:03

Just choose some expression for $F$ and solve for $G$. $F(x)=x^2$ makes the things easy, for example: $$x=\sqrt{x^2+G(x)(x+n)}-(x+n)$$ $$(2x+n)^2=x^2+G(x)(x+n)$$ $$G(x)=\frac{3x^2+4xn+n^2}{x+n}=3x+n$$

  • $\begingroup$ But this still should be for $x>-n$, otherwise you should put absolute values of $(x+n)$ $\endgroup$ – Svetoslav Jul 29 '15 at 14:15

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