# Definition of a unique function

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.

## Answers and Replies

tiny-tim
Homework Helper
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

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

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