A function corresponding to $(0,0)$, $(1,1)$, $(1+\frac12,2)$, $(1+\frac12+\frac14,3)$, $(1+\frac12+\frac14+\frac18, 4)$, etc

Question

Earlier today I decided I wanted to see if I could make a function that would correspond to the following:

$$\begin{array}{c|c} x & y \\ \hline 0 & 0 \\ 1 & 1 \\ 1+\frac12 & 2 \\ 1+\frac12+\frac14 & 3 \\ 1+\frac12+\frac14+\frac18 & 4 \\ \end{array}$$ and on and on...

So, to make a long story short, I have ended up putting functions in functions to end up with the right numbers in the $n$-th terms. It seems to never end. Surely I'm missing something. I'd be thrilled if anyone can fill me in.

Answer 1

If you want $x$ as a function of $y$ you can use $$x=2-2^{1-y}$$ If you want $y$ as a function of $x$ you can use $$y=-\log_2(2-x)+1$$