Definition of a unique function

by jonsploder
Tags: definition, function, unique
 P: 2 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.
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P: 26,167
hi jonsploder! welcome to pf!
 Quote by jonsploder … Are these two f(x)'s the same function, or different functions …
they're different

they're both restrictions of the same function defined on the whole of R
 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.
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P: 26,167

Definition of a unique function

 Quote by jonsploder 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.
they're functions, and they're different

so they're different functions

the definition of a function includes its range and domain: different range and/or domain, different functions
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P: 775
 Quote by jonsploder 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.
The domain and codomain are part of the definition of a function.

Two functions $f : A \to B$ and $g : C \to D$ are equal if and only if $A = C$ and $B = D$ and for all $a \in A$, $f(a) = g(a)$.
P: 210
 Quote by pasmith Two functions $f : A \to B$ and $g : C \to D$ are equal if and only if $A = C$ and $B = D$ and for all $a \in A$, $f(a) = g(a)$.
This definition is exactly correct. That should be your definition.

....

It's worth noting, however, that sometimes people get lazy about codomains and say $f : A \to B$ and $g : C \to D$ are equal when $A = C$ and for all $a \in A$, $f(a) = g(a) \in B\cap D$.
P: 779
 Quote by economicsnerd It's worth noting, however, that sometimes people get lazy about codomains and say $f : A \to B$ and $g : C \to D$ are equal when $A = C$ and for all $a \in A$, $f(a) = g(a) \in B\cap D$.
To explain why this definition is bad, consider
##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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