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I assume that, when a math book says a function is greater than the other, i.e. $f > g$ where $f$ and $g$ are functions, $f$ is greater than the $g$ in the pointwise, everywhere on the domain of these functions. So I also assume that, domain of these functions should agree. Am I right? Or what are the other implications of this statement?

Thanks.

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    Your interpretation seems quite right (with range instead of somain - typo?), but it might depend on the context. Some author might write f>g e.g. for functions defined on $\mathbb N$ if we have f(n)>g(n) for almost all $n$. In other contexts, one might write f>g if the domain of $g$ is a strict subset of the domain of $f$ and $g$ is the restriction of $f$ to that domain.2012-11-07

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If the domain of $f$ is the set $F$ and the domain of the function $g$ is $G$, then a typical interpretation of $f>g$ is: $\text{For all $x \in F \cap G$, we have $f(x) > g(x)$.}$ It is not necessary that the range of the functions should agree. For instance, $f(x) = 1$ has range $\{1\}$, whereas $g(x) = 1-e^{-x}$ has range $(-\infty,1)$.