Questions about what is an onto function and what is not

[relinquished™]

onto or not?

Hello ^_^

I just have a few questions regarding onto functions. I'm a student studying BS Math here in the Phils. Right now i have a subject concerning math logic, and before we study the subject proper, my professor is discussing the basics of relations an functions ^_^;;; So if this question is misplaced I am truly sorry ^^;;;

In my previous quiz, there was a question that asked "State whether the given function is onto or not:

Domain = [-4,4], f(x) = x^2
Domain = [-1,1], f(x) = sin x

My understanding of an onto function is that it is a function wherein all the members of the codomain of the function should be assigned to at least one value of x in the domain. I answered that "yes, f(x)=x^2 is an onto function" because all the values of x in the domain have a corresponding y, i.e., there is no undefined value for any value of x or y. The next question's answer was was the same. However, when I asked for clarifications on what were the correct answers, my professor said that these two functions were not onto. Can someone please tell me why?

Thanx in advance ^_^

 

[AKG]

I'm a student studying BS Math here

Domain = [-4,4], f(x) = x^2
Domain = [-1,1], f(x) = sin x

My understanding of an onto function is that it is a function wherein all the members of the codomain of the function should be assigned to at least one value of x in the domain. I answered that "yes, f(x)=x^2 is an onto function" because all the values of x in the domain have a corresponding y, i.e., there is no undefined value for any value of x or y. The next question's answer was was the same. However, when I asked for clarifications on what were the correct answers, my professor said that these two functions were not onto. Can someone please tell me why?

Thanx in advance ^_^

Check out this Wikipedia article. Based on what's in that article, I don't understand how you can answer that function either way because there is no co-domain specified. The answer is yes to the first question if and only if the co-domain is a subset of [0,16]. The answer to the second question is yes if and only if the co-domain is a subset of [sin(-1), sin(1)].

 

[relinquished™]

Thanks for the clarifications ^_^

 

[DrMatrix]

Check out this Wikipedia article. Based on what's in that article, I don't understand how you can answer that function either way because there is no co-domain specified. The answer is yes to the first question if and only if the co-domain is a subset of [0,16]. The answer to the second question is yes if and only if the co-domain is a subset of [sin(-1), sin(1)].

The co-domain for the function in the first question cannot be a proper subset of [0,16] or you would not have a function. Where you said "is a subset of" should be replaced with "equals" for both questions.

 

[gravenewworld]

Yes I don't see how you can answer that question. The values that the function maps to must be specified in order for you to answer the question.

 

[AKG]

The co-domain for the function in the first question cannot be a proper subset of [0,16] or you would not have a function. Where you said "is a subset of" should be replaced with "equals" for both questions.

What makes you say this? It's definitely a function. Check out that wikipedia link for what an "onto function" is for clarification. I believe I understood it correctly.

 

[Muzza]

I don't think it has anything to do with the function being onto or not. Can you exhibit a proper subset S of [0, 16] such that f: [-4, 4] -> S, f(x) = x^2 is a function? Remember,

Formal definition
...
1. f is total: for all x in X, there exists a y in Y such that x f y (x is f-related to y), i.e. for each input value, there is at least one output value in Y.

So (for example) f: [-4, 4] -> [0, 4] won't work since then f(3) = 9 would be in [0, 4].

 

[AKG]

Muzza you're right.