Register to reply

Function Co-domain!

by dijkarte
Tags: codomain, function
Share this thread:
dijkarte
#1
Apr6-12, 01:23 PM
P: 200
Given a function f(x) f:A --> B, can the choice of codomain affect whether or not the function is surjective? For instance, f(x) = exp(x), f:R --> R is an injection but not surjection. However, assuming we can vary the co-domain, and lets make it f: R --> (0, inf), f(x) is now bijection. Is this correct?
Phys.Org News Partner Mathematics news on Phys.org
'Moral victories' might spare you from losing again
Fair cake cutting gets its own algorithm
Effort to model Facebook yields key to famous math problem (and a prize)
Number Nine
#2
Apr6-12, 02:16 PM
P: 772
Yes. The codomain pretty much determines by itself whether or not the function is surjective.
dijkarte
#3
Apr6-12, 02:24 PM
P: 200
Got it. Thanks.

Number Nine
#4
Apr6-12, 04:23 PM
P: 772
Function Co-domain!

Quote Quote by dijkarte View Post
Got it. Thanks.
I should probably qualify my previous post. The codomain determines surjectivity in the sense that if you define the codomain to be the same as the range of the function, then your function becomes surjective. In fact, given any function, you can restrict the codomain in such a way as to make the function surjective (you can also restrict the domain in such a way to make it injective). A different function with the same codomain obviously may not be surjective.
dijkarte
#5
Apr6-12, 06:07 PM
P: 200
But there's no rule which restricts the specification of the codomain based on the mapping rule itself as long as the range is a subset of the codomain. For instance, f(x) = sin(x) can be specified as:

f: R --> R
f: R --> [-2, 2)
f: R --> [-1, 1]

but not as f: R --> [0, 4]


Register to reply

Related Discussions
Domain of a function Precalculus Mathematics Homework 4
Domain of a Function Calculus & Beyond Homework 2
Domain of a function Precalculus Mathematics Homework 3
Domain of a function General Math 7
What is the domain of this function? General Math 11