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It's easy to solve for $x$ in this equation: $$y=x-f(x)-1$$ where $f(x)$ is a different function than y, but I need to solve for $x$ without $f(x)$ on the other side. How would I accomplish this? Also, does it matter the technique used to solve this what $f(x)$ is?

If it is not possible without knowing what the function is, for my uses it is a version of $\pi(x)$, the prime counting function, only defined for composites. So, that is, what is the inverse of the function $x-f(x)-1$ where $f(x)$ is $\pi(x)$ only defined for composite numbers?

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  • $\begingroup$ Unless you just want $x-f(x) = y+1$, I don't think what you're asking for can be done. There's no way to get to rid of the $f(x)$ (i.e. combine it with the other terms and simplify) without knowing explicitly what it is. $\endgroup$ – user137731 Jun 17 '16 at 23:42
  • $\begingroup$ Well, if you would like to know, for my uses, it is $\pi(x)$, the prime counting function. $\endgroup$ – asher drummond Jun 17 '16 at 23:47
  • $\begingroup$ There's nothing you can do to get rid of that $f(x)$ with just simple algebra. Depending on what $y$ and $f$ are, you might get some useful cancellation in $y-f(x)$, but there's certainly no general method that works for all $f$. $\endgroup$ – Jack M Jun 17 '16 at 23:53
  • $\begingroup$ The function $ f(x) =x - \pi (x) - 1 $ has no inverse, since $ f(2) = f(3) = 1 $. $\endgroup$ – cardboard_box Jun 17 '16 at 23:59
  • $\begingroup$ Ah, I forgot something. I will modify the answer. $\endgroup$ – asher drummond Jun 18 '16 at 0:01

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