Definition of a unique function

[jonsploder]

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.

 

[tiny-tim]

hi jonsploder! welcome to pf!

… 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

 

[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.

 

[tiny-tim]

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

 

[pasmith]

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 $a \in A$, $f(a) = g(a)$.

 

[economicsnerd]

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 $a \in A$, $f(a) = g(a) \in B\cap D$.

 

[pwsnafu]

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.