<s>[INST] How do we write $f(x+2)$ in terms of $f(x)$? $$f : R^+ \rightarrow R^+$$
$$f(x)  = \dfrac{x}{x+1} $$
How do we write $f(x+2)$ in terms of $f(x)$? This is a general question that I wondered how to algebraically write it. 
Regards
 [/INST] A more general method:
$$f(x) = \frac{x}{x+1}$$
$$f(x)(x+1) = x$$
$$(f(x)-1)x=-f(x)$$
$$x=\frac{f(x)}{1-f(x)}$$
Hence,
$$f(x+2)=\frac{x+2}{x+3}=\frac{\frac{f(x)}{1-f(x)}+2}{\frac{f(x)}{1-f(x)}+3}=\frac{f(x)+2(1-f(x))}{f(x)+3(1-f(x))}=\frac{2-f(x)}{3-2f(x)}$$
Note: I usually think that this can easily go wrong, so you may want to do some checking: $f(1+2)=\frac{3}{4}$ by the original expression, and indeed $f(1+2)=\frac{2-f(1)}{3-2f(1)}=\frac{2-\frac{1}{2}}{3-2(\frac{1}{2})}=\frac{\frac{3}{2}}{2}=\frac{3}{4}$.
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