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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1There are a lot of different notions of equivalent for functions. The notion you've defined is equality, not equivalence. – Thomas Andrews Jan 24 '14 at 02:11
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1In what context did you encounter this? The correct interpretation may be context-dependent. – Cameron Buie Jan 24 '14 at 02:13
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If you give the same input to both, you get the same output from both. So, yes, they are equivalent from that perspective. – Dan Christensen Jan 24 '14 at 04:31
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1Re: Hold... Jeez, guys! What could be more clear??? – Dan Christensen Jan 24 '14 at 17:48
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Sticklers may object to your usage of the equals sign in $f=g$. Maybe you should use $f\equiv g$. – Dan Christensen Jan 24 '14 at 20:32
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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.
ncmathsadist
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1Wouldn't the first and third statements imply the second? After all, if for all $x$, $f(x) = g(x)$, then it follows that every element of the range of $f$ is also in the range of $g$, and conversely. – Jan 24 '14 at 03:59
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4@Strants I'm thinking ncmathsadist meant codomain instead of range. – Cameron Williams Jan 24 '14 at 04:03
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I make a distinction. There is the image of a function; this is $f(D)$, where $D$ is the domain of $f$. The range of a function is the set you are specifying it takes values in. If $f:[0,\infty)\to \mathbb{R}$ is defined by $f(x) = x$ and $g:[0,\infty)\to [0,\infty)$ is defined by $g(x) = x$, I see these as different functions. – ncmathsadist Jan 24 '14 at 23:01