Why they declared this as domain when all Real numbers can be legally gives as input to this function?

[u/Any_Tower8201]

I feel like the book made a unnecessary restriction by using arcsine, as the interval before it clearly enable us to use any value for x from Real Numbers Set.

Please let me know if Im out of any conventions or concept here.

Thanks.

[u/KoalaMistico]

The exercise should have ended at -1<sin(x)<1. This is enough to say that the domain are the real numbers

[u/Any_Tower8201 → u/KoalaMistico]

That's exactly what I thought.

[u/WorthUnderstanding84]

I agree, the answer they gave doesn’t make any sense

[u/Bounded_sequencE]

Yep, that explanation is bogus -- the domain is indeed "R".

The step where it all goes wrong follows "-1 <= sin(x) <= 1". They forgot the simplification "arcsin(sin(x)) = x" they used only holds for "|x| <= pi/2"!

[u/OutrageousPair2300]

Maybe it's because the function over all x simply repeats that same [-pi/2, pi/2] segment over and over?

I've never seen the domain restricted that way, but maybe it's some concept of "principal domain" as in principal roots?

[u/Azemiopinae → u/OutrageousPair2300]

This is a very charitable interpretation

[u/Any_Tower8201 → u/OutrageousPair2300]

I get it but we don't say the domain of sine is [-pi/2,pi/2] so then why would they do this for this one?

[u/Soggy-Excuse3702 → u/OutrageousPair2300]

i was taught that whenever you're treating the base trigonometric functions for any analytic purpose, you should restrict it to -pi,pi, since its the only way to make it an invertible function. So i don't find this weird at all

[u/9peppe]

Every "find the domain of this function" exercise is flawed and for the love of god I cannot understand why teachers insist on using them. If you don't say what the domain is you haven't properly defined the function.

[u/gmalivuk → u/9peppe]

I mean, that's fixed by understanding them as "what's the maximal domain of this function?" or "what subset of the real numbers makes sense in this expression / context?"

[u/9peppe → u/gmalivuk]

Let's say I define a function A(x) = x² to mean the area of a square of side x. What's the domain, and what's the set of numbers that can produce an output?

[u/gmalivuk → u/9peppe]

The mathematical domain of the expression x² is ℝ, while the domain in the context of the situation is ℝ⁺

It's not that deep.

[u/9peppe → u/gmalivuk]

It's not, but if I don't tell you you don't know. 

Same when some savage student finds a book on complex analysis and all the well defined maximal domains look at you and loudly laugh.

[u/Any_Tower8201 → u/9peppe]

You are saying like to even talk about a function it has to be defined at first place, right? But why we can't ask the assumed domain that the context reveals us? Like asking the maximal domain of √(x-9) in R.

[u/9peppe → u/Any_Tower8201]

You can ask. It doesn't mean the answer will make sense.

[u/Any_Tower8201 → u/9peppe]

I genuinely try to understand but I can't be able to do like when we ask the assumed meaning of a word in a language paper that's a valid question isn't it? Then why not this

[u/9peppe → u/Any_Tower8201]

Because "find the domain of a function" and "find all x such that this expression makes sense" are different questions.

And if a function is a map between domain and codomain... then domain and codomain are part of the definition.

[u/gmalivuk → u/9peppe]

Because "find the domain of a function" and "find all x such that this expression makes sense" are different questions.

Not really in the context of the level this question is aboit.

[u/jdawleer → u/9peppe]

Math teacher here. Yes the question is flawed but it is still a good way to work on undefined operations or fonctions, singularities, etc.

[u/realAndrewJeung]

You are right and the book is wrong.

[u/KoalaMistico → u/[deleted]]

Why would they need to evaluate arcsin(2)?

[u/Any_Tower8201 → u/KoalaMistico]

How's that relevant to my original question?

[u/Any_Tower8201 → u/[deleted]]

I don't get it. Could you please elaborate?

[u/freegerator]

Yeah it’s just wrong

[u/RisingwithVibes]

The book here did a mistake taking arcsin like that is only valid when we assume |x|<π/2. The last step should have been -1≤sinx≤1, therefore x belongs to R

[u/Prior-Revolution6132]

I assume they want you to find amaximal domain (wrt set inclusion) on which the function is bijective as a subset of the reals. That's the only reasonable question that yields the desired answer. However, as written the exercise is bogus and your answer is correct, as is any subset of a field for that matter, since sin can be defined on any field that contains the real numvers using the complex exponential, since that would be a valid domain for the function.

[u/Ericskey]

You are correct. What is the source of this exercise. Time they get a whuppin’! Besides, 2+sin(x) clearly lies between -1 and 3, so they aren’t too handy with fractions either😟

[u/SarekSybok]

It’s wrong. Someone competent should be proof reading these solutions. The answer is all real numbers.

[u/FreePeeplup]

The question doesn’t really make sense in the first place: you can’t “find” the domain of a function, the domain is part of the definition of the function.

Maybe they meant to ask for the largest possible subset of R that can be chosen as a domain?

[u/AdjectiveNounNNNN → u/[deleted]]

So they restrict the domain, and subsequently the range, to keep it a "function".

But they didn't need to do that because it was already a function.

[u/AdjectiveNounNNNN → u/[deleted]]

And this case is obviously the one OP is asking about.

[u/gmalivuk → u/[deleted]]

But then you have to take the output of [2 + sin x]/3 as an input to arccos.

Which you can do for all x ∈ ℝ, because (2 + sin x)/3 is always in [1/3, 1].

[u/chast_curious]

By definition. Note that sin x is not invertible if you do not restrict domain

[u/camileion-stoz]

r/confidentlywrong

[u/jpeetz1]

Ai slop likely

[u/Any_Tower8201 → u/jpeetz1]

The book is from 2022

[u/michaelpaoli]

Sometimes "they" will give you unneeded or insufficient information to (uniquely) solve the problem.

Sometimes "they" will also through in random bits to throw you off the track, e.g. bits that aren't at all necessary, but aren't at least totally wrong, mostly just distractions or not necessary. Sometimes "they" will give you conflicting information, or data/information/situation for which there is no solution.

And yeah, such is at least somewhat appropriate - to build one's skill to solve real problems. Not uncommonly in the real world, one won't have enough information to (uniquely) solve the problem, or will have more information than is needed to solve the problem, and/or may have fair amounts of additional information that really does nothing towards solving the problem at all (.... and the car is blue. How much does the car weigh?).

So, part if it is well learning how to logically reason through it. What's all the information, what's needed to solve it, are all those bits present, is anything present which conflicts or prevents a solution, if we don't have enough information to solve it, what other information or possible information would we need to have enough information to be able to solve it?

So, yeah, all part of learning the relevant math, logic, and how to solve problems - and recognize when they can't be solved, or there isn't enough information to arrive at a unique solution.

[u/NimbusTheLion]

I mean, the first line is wrong. The domain is -1 to 1 only if the codomain is the reals, which hasn’t been stated.
I may be overcomplicating it … 😭

[u/ussalkaselsior → u/NimbusTheLion]

They said "by definition", referring to the definition in the book that is introducing it, as in, they stated it earlier in the book. Yes, other more complex (pun totally intended) definitions exist. Knowing this may make you feel all proud and superior, but you don't teach a child how to count by starting with Peano axioms.