# What does it mean for a function to be polynomially bounded

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

If a function $f$ is polynomially bounded it means there exists polynomials $g$ and $h$ such that for all $x$, $$g(x)\le f(x)\le h(x).$$
• E.g. $e^x$. Remember the inequality has to hold for all $x$. Mar 20 '13 at 23:34
• @bigO: $x\mapsto 2^x$ is not polynomially bounded. Mar 20 '13 at 23:34
• @bigO: $3^x$ is not a polynomial! Mar 20 '13 at 23:37