If I had the table of $x$ , $f(x)$ pairs, and this is the rule:
\begin{array} {|r|r|} \hline 1 &1 \\ \hline \hline x_i & f(x_i) \\ \hline \hline 2x_i & 1 +f(x_i) \\ \hline \end{array}
Basically that means that for every $x$ the function value will increment at the $2x$ point.
I'm struggling to find out the exact function, in which i can input the $x$, and it will give me the correct output. :)
I know that the function will look something like this.
So far I constructed this table \begin{array} {|r|r|} \hline x & f(x) \\ \hline \hline 1 &1 \\ \hline \hline 2 &2 \\ \hline \hline 3 & k \\ \hline \hline 4 & 3 \\ \hline \hline 5 & \\ \hline \hline 6 & k+1 \\ \hline \hline ... & ...\\ \hline \hline 8 & 4\\ \hline \hline ... & ...\\ \hline \hline 12 & k+2\\ \hline \hline ... & ...\\ \hline \hline 16 & 5\\ \hline \hline ... & ...\\ \hline \end{array}
where $k$ is the number between 2 and 3, obviously.
I tried interpolating that table in wolfram, but it can't work. I think that there is simple solution to this function. Or is there?
Feel free to edit the question if you think that it can be improved.