MathLabs

Problem 5

Determine all functions f:R→Rf:\mathbb R\to\mathbb R such that (1) ff is strictly increasing, and (2) f(x)+g(x)=2xf(x)+g(x)=2x for every real xx, where gg is the composition inverse of ff.
Step 1 of 5: Apply the equation to an iterate of f
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fn+2(x)+fn(x)=2fn+1(x)f_{n+2}(x)+f_n(x)=2f_{n+1}(x)
Detailed analysis

Let fnf_n denote the nn-th iterate, with f0(x)=xf_0(x)=x. Substitute x=fn+1(x)x=f_{n+1}(x) into f(x)+g(x)=2xf(x)+g(x)=2x. Since g(fn+1(x))=fn(x)g(f_{n+1}(x))=f_n(x), the result is fn+2(x)+fn(x)=2fn+1(x)f_{n+2}(x)+f_n(x)=2f_{n+1}(x).