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I have two functions: $f(x) = x^{2}-3|x-1|$ and $g(x)=2|x-2|$.

I need to find the sum of all integer solutions for the following inequality: $$g[f(x)]\leq 2$$

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    $\begingroup$ Do you mean by $[.]$ the floor function or it is just parentheses, $g(f(x))$? $\endgroup$ – mrs Nov 26 '12 at 9:10
  • $\begingroup$ It was not specified in the task, but I guess it's a rather a floor function. $\endgroup$ – kjempeflaux Nov 26 '12 at 9:16
  • $\begingroup$ The floor function would make no sense, both because $f$ takes integer values at the integers anyway, and because that would leave $g$ without an argument; so it must be intended as $g(f(x))$. $\endgroup$ – joriki Nov 26 '12 at 10:30
  • $\begingroup$ I get it now. Thanks, @joriki. And so it goes as g(f(x)). $\endgroup$ – kjempeflaux Nov 26 '12 at 10:38
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Presumably by "roots" you mean solutions. (The term "roots" is usually reserved for solutions of equations.)

The inequality is fulfilled if $|f(x)-2|\le1$. Since $f(x)$ takes integer values at the integers, that means $f(x)\in\{1,2,3\}$. Substituting the integers from $-5$ to $5$ shows that the only solution is $x=3$.

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  • $\begingroup$ That was really helpful, cheers! $\endgroup$ – kjempeflaux Nov 26 '12 at 15:28
  • $\begingroup$ @Max: You're welcome! $\endgroup$ – joriki Nov 26 '12 at 15:28

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