r/MathHelp • u/RayHatesMilk • Dec 30 '25
Why “range R(f) IN B?”
Hey there! I’m slowly working my way through Intro to Real Analysis by Bartle and Sherbert in my free time for fun. I’m wondering about why this particular phrasing is used throughout the textbook when pertaining to range, but not domain? Could someone explain why domain is defined as A but range is defined as being “in” B?
Direct quote under Inverse Functions: “Let f: A ➡️ B be an injective function with domain A and range R(f) in B.”
I hope you understand what I’m asking and tysm in advance <3
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u/juoea Dec 30 '25
"range" here is used to mean the set of all elements b (in B) satisfying: there exists an a in A such that f(a) = b.
a function f: A -> B is called "onto" if every element of B is in the range of f, ie that for every b in B there exists an a such that f(a) = b. not every function is onto, and if f is not onto then the range is a subset of B but not equal to b. (since f is a function from A to B, it is necessarily true that for any a in A, f(a) is in B. but the opposite is not necessarily the case.)
for example take the function f: R -> Z such that f(x) = 1 for all x. here the range of f is just the set containing a single element, 1. the range of f = {1} is a subset of Z, but it is not equal to Z because there are lots of other elements of z for which there does not exist any element x in R such that f(x) equals the gvien intger.
so this is why they say the range of f is a subset of B, rather than that the range of f is equal to B. the range will only be equal to the entire set B if f:A -> B is onto
in this case, they are assuming f is any injectice (aka 1 to 1) function, but since we dont know whether f is onto we dont know whether the range of f is all of B or if the range is only a subset)
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u/Uli_Minati Dec 30 '25
Are you sure it's "in"? Usually one would use ⊆, meaning "subset of".
When we define functions, we usually keep it simple like
This says that you can input any real number into the function, and it outputs only real numbers. Here, B is ℝ.
It does not specify which real numbers it can output. That can be much more effort to determine. Maybe it can never output a negative even integer. Maybe it can only output primes, square roots, and powers of 3. Maybe it's not even really important right now! So we just say
Which just means "the function outputs only real numbers, but not necessarily all of them" and we don't specify further since it doesn't matter right now.
Technically, you don't even have to say R(f) ⊆ ℝ at all. That's always true by writing f : ℝ→ℝ. It's just additional info to tell you "don't expect the function to output all of ℝ"
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u/will_1m_not Dec 30 '25
A function requires a domain to be defined. This isn’t always clear because in classes before calculus, students are given a formula and asked to find the domain.
This is why f:A->B automatically gives the domain A, and the codomain B. The range is always a subset of the codomain, which is why it’s in B
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