# Find all functions $f:(0,\infty)\rightarrow(0,\infty)$ subject to the conditions: [duplicate]

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Find all functions $$f:(0,\infty)\rightarrow(0,\infty)$$ subject to the conditions:

$$f(f(f(x)))+2x=f(3x)$$ for all $$x>0$$ and $$\displaystyle\lim_{x\to\infty}(f(x)-x)=0$$

I tried as follows: Suppose $$x_0=x,x_1=f(x_0)=f(x),...,x_n=f(x_{n-1})$$. But I am facing problem due to $$f(3x)$$ term.

## marked as duplicate by Sil, Adrian Keister, Community♦May 30 at 18:41

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## 1 Answer

Note that $$f(x)=x$$ satisfies all three of your conditions, by inspection.

• Yes. But How to prove it mathematically. – J.Doe May 30 at 18:18
• There's almost nothing to prove. $\displaystyle\lim_{x\to\infty}(f(x)-x)=\lim_{x\to\infty}(x-x)=\lim_{x\to\infty}0=0.$ Then $f(f(f(x)))+2x=f(f(x))+2x=f(x)+2x=x+2x=3x=f(3x).$ Finally, note that if $x>0,$ it follows that $f(x)=x>0,$ so that $f:(0,\infty)\to(0,\infty).$ Done. – Adrian Keister May 30 at 18:20
• Oh, but you mean show that there aren't any other solutions. That's a lot harder. Not sure I know how to do that. Whatever function you had would certainly need to approach $f(x)=x$ asymptotically (the limit condition means you have a slant asymptote). – Adrian Keister May 30 at 18:21