What does it mean rigorously for two functions $f: \mathbb{R} \to \mathbb{R}$ and $g: \mathbb{R} \to \mathbb{R}$ to be equivalent? Does $f = g$ if and only if $\forall x \in \mathbb{R} \ \ f(x) = g(x)$?
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$\begingroup$It means three things. First, the domains of the two functions must be the same. Secondly, the ranges (as apposed to images) of the functions must be the same. Thirdly, for each element of the domain, the rule of the two functions must yield the same result.
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