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.
hi jonsploder! welcome to pf! 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.
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
The domain and codomain are part of the definition of a function. 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].
This definition is exactly correct. That should be your definition. .... 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].
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.