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There is a definition in my notes and says,

When functions are polynomially bounded, the initial conditions (the value on small inputs) do not make a difference for the solution in asymptotic notations. The initial conditions can make a difference, when the function is not polynomially bounded.

Can somebody explain what that means? Thank you

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If a function $f$ is polynomially bounded it means there exists polynomials $g$ and $h$ such that for all $x$, $$g(x)\le f(x)\le h(x).$$

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  • $\begingroup$ Isn't there always a function h(x) that is greater than f(x)? Can you give an example of a polynomially unbounded function? $\endgroup$ Mar 20 '13 at 23:32
  • $\begingroup$ E.g. $e^x$. Remember the inequality has to hold for all $x$. $\endgroup$ Mar 20 '13 at 23:34
  • $\begingroup$ @bigO: $x\mapsto 2^x$ is not polynomially bounded. $\endgroup$ Mar 20 '13 at 23:34
  • $\begingroup$ @HenningMakholm forr example, if we choose h(x)=3^x, doesn't the inequality hold? I couldnt get the point $\endgroup$ Mar 20 '13 at 23:37
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    $\begingroup$ @bigO: $3^x$ is not a polynomial! $\endgroup$ Mar 20 '13 at 23:37

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