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Definition of a unique function 
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#1
Jan2714, 04:12 AM

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Hi all, I'm wondering whether an expression which is used to describe a function in a certain domain is a different function for the same expression with a differing domain.
For example: expression; x^2. f(x) = x^2 for domain {1 < x < 10} f(x) = x^2 for domain {10 < x < 11} Are these two f(x)'s the same function, or different functions, by definition. I couldn't be sure by Wikipedia, and it's a difficult question to type into a search engine. 


#2
Jan2714, 05:23 AM

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hi jonsploder! welcome to pf!
they're both restrictions of the same function defined on the whole of R 


#3
Jan2714, 05:33 AM

P: 2

Thanks for the welcome, and the reply.
I know that they are different, however I was wondering, by the most formal definition of a function, whether they are different functions, or if indeed the domain of a function constitutes its identity as a function. 


#4
Jan2714, 05:42 AM

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Definition of a unique function
so they're different functions the definition of a function includes its range and domain: different range and/or domain, different functions 


#5
Jan2714, 06:31 AM

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Two functions [itex]f : A \to B[/itex] and [itex]g : C \to D[/itex] are equal if and only if [itex]A = C[/itex] and [itex]B = D[/itex] and for all [itex]a \in A[/itex], [itex]f(a) = g(a)[/itex]. 


#6
Jan2714, 06:02 PM

P: 237

.... It's worth noting, however, that sometimes people get lazy about codomains and say [itex]f : A \to B[/itex] and [itex]g : C \to D[/itex] are equal when [itex]A = C[/itex] and for all [itex]a \in A[/itex], [itex]f(a) = g(a) \in B\cap D[/itex]. 


#7
Jan2714, 06:33 PM

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##f : \mathbb{R} \to \mathbb{R}##, ##f(x) = 0## ##g : \mathbb{R} \to \{0\}##, ##g(x) = 0##. Note that under the definition economicnerd gave these would be considered equal. However, g is a surjection while f is not. 


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