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What I figured is that it means that the function has an upper bound, however I came across this text:enter image description here

Here since g(x) either equal or less to f(x), |g(x) / f(x)| must be bounded right? Since the denominator is greater than the numerator in this case, the value of the fraction can't exceed a lot.

What am I interpreting wrong here?

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Bounded means bounded above and below. The function $-x^2$ is not bounded. In most applications to algorithms, it doesn't matter, since the functions are naturally non-negative, so are bounded below by $0$. But in principle one should be careful. The function $f(x)$ is bounded if and only if $|f(x)|$ is bounded above. – André Nicolas Feb 10 '13 at 5:43
Have you looked at in depth? – JB King Feb 10 '13 at 5:44

A function $f: D \to \mathbb{R}$ is said to be bounded if there exists a constant $C>0$ such that $|f(x)| \le C$ for all $x\in D$. So it is both lower- and upper- bounded. And $C$ need not be less than $1$.

The big-$O$ notation gives something else, called asymptotic bound. Don't mix up.

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Indeed, if $|f(x)/g(x)|$ is bounded, so is $|g(x)/f(x)|$. If one bound is $C$, the other is $1/C$. – Laura Balzano Feb 10 '13 at 5:52
Not really. See what if $f(n)=n$ and $g(n) = n^2$ where $n\in\mathbb{N}^*$. – Robin Feb 10 '13 at 5:57
Oops, Thank you @Robin! – Laura Balzano May 21 '13 at 20:58

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