Matematické Fórum

stuart clark

If $f(x)$ be a periodic function and $g(x)$ be a non periodic function and $f(g(x))$ is periodic function

and $g(2) = 3$ and $g(4)=7,$ Then $g(6)=$

check_drummer · 20. 03. 2016 22:12

↑ stuart clark:
Hi, my guess is 11 (for g to be linear) but it remains to prove (along with that the guess is true) that other values are forbidden...

Brano · 26. 03. 2016 19:53 — Editoval Brano (26. 03. 2016 19:57)

the problem is not sufficiently described. As is the answer could be anything - even with some minor additional assuptions like continuity of functions.

let $\phi$ be a any continuous function s.t. $\phi(0)=0$ and $\phi(1)=1$ and let $g(x)=T\phi$\{\frac{x}{T}\}$+T\[\frac{x}{T}\]$ where $T>0$ and $\{\}$ denotes fractional part and $[]$ denotes integer part.
Then for any T-periodic function $f$ , the function $f\circ g$ is T-periodic.

Now you can play with parameters $T,\phi$ to obtain different values of $g(6)$ ; e.g. fix $T=7$ and find suitable $\phi$