# Best approximation of a real number with two functions

There's two functions, called $$F(x)$$ and $$G(x)$$, where $$F(x)>x,1 ,$$F'(x)>0, G'(x)>0$$ on $$(2,\infty)$$, and $$F(x),G(x)\in(2,\infty)$$. Given $$x, y\in (2,\infty)$$, and now I want to find the best way to approximate $$y$$ in finite steps, that is, $$F(G(G(F(x)...)))\sim y$$.

For example, given $$F(x)=\Gamma(x+1),G(x)=\sqrt{x}, x=8, y = 2.2$$, and set the maximum steps to $$3$$. It can be proved that the best approximation is $$\sqrt{\Gamma \left(\sqrt{8}+1\right)}=2.20394...$$

It could be easily done with BFS, however I think there should be a faster way to give the result.

Meanwhile, I haven't found any results about the best approximation (e.g. the estimation of the error of the best approximation)

Does anyone have ideas?

• Do you know anything more about F and G? E.g. are they both strictly increasing? – Peter Taylor Aug 10 at 15:07
• Without more info, this problem is not answerable. In particular, if $g(x)=0$, then the problem becomes degenerate. – Don Thousand Aug 10 at 15:10
• Oh yeah they are both increasing. thx – FFjet Aug 10 at 15:10
• why $F(x)>x-1$ ? – G Cab Aug 10 at 15:19
• well I have just modified the conditions – FFjet Aug 10 at 15:23