
#1
Jan2714, 04:12 AM

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. 



#2
Jan2714, 05:23 AM

Sci Advisor
HW Helper
Thanks
P: 26,167

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

Sci Advisor
HW Helper
Thanks
P: 26,167

Definition of a unique functionso 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

HW Helper
P: 772

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: 206

.... 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

Sci Advisor
P: 773

##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. 


Register to reply 
Related Discussions  
Show two unique points lie on a unique line.  Calculus & Beyond Homework  6  
Linear Algebra, unique minimizer of a quadratic function  Calculus & Beyond Homework  1  
f >= g function definition  Topology and Analysis  2  
Number theory proof: Unique determination of a recursively defined function  Calculus & Beyond Homework  5  
What does it mean for a function to be unique?  Calculus  3 