Set Theory: prove f(x)= x ÷ sqrt[x^2+1] is Bijec. on (-1, 1)

[itscomplicated]

Was hoping I could get some help showing the steps of this problem.

Prove that the following is a bijection.
range (-1,1) f(x)= x ÷ sqrt [ (x^2) + 1 ]

[nona.m.nona]

Prove that the following is a bijection.
range (-1,1) f(x)= x ÷ sqrt [ (x^2) + 1 ]

A good place to start is with the definition. What is the definition of a "bijection"?

When you reply, please include a clear listing of your efforts so far. Thank you.

[itscomplicated]

Definition of a Bijection is that the function must be Injective AND surjective and therefore it is Bijective. Did you need to know the definition or are you asking if I even know it? as for my steps I am not quite clear how to approach it and the only help I have gotten so far is that since it is a strictly increasing function it is therefore Injective. but how to prove it more than that I do not know. Of course after you must prove that it is Surjective. If you need the definitions of those I can provide them for you.

[nona.m.nona]

as for my steps I am not quite clear how to approach it....

That's odd. Your book and your class were supposed to have covered this material before having assigned exercises on it. (Also, the way this function is defined is quite odd, since you are given the range of the function, rather than the domain.)

Since you are not familiar with the general methodology, hints and suggestions on how to use that methodology obviously won't provide you with any intelligible assistance. Therefore, it would appear that the first step will be for you to learn the information which has, for some unfathomable reason, thus-far been withheld from you. The following may be helpful:

Injective, Surjective, and Bijective
Functions (PDF)
Functions Review (PDF)
Properties of Functions

Note in particular that, to prove surjectivity (that is, that the function is "onto"), the customary process is to pick some random element in the target set (the listed range, one supposes) and show that there is some element in the domain which maps onto that range element.

[itscomplicated]

The book the teacher assigned to my class simply is a different method of approaching proofs than most (it literally says it). Anyways one of the ways to approach the problem was to say, Assume there is an x sub 1 in f(x) and another x sub 2 in f(x) you can then prove they are equal thus proving that x sub 1 is = to x sub 2 and therefore proving x = y. (does not completely make sense but thats what I have been given. So I take that and what I have learned in class and have 2 methods learned.

{(xsub1) / sqrt[ (xsub1^2) +1] } = {(xsub2) / sqrt[ (xsub2^2) +1] } ---> {(xsub2) * sqrt[ (xsub1^2) +1] } = {(xsub1) * sqrt[ (xsub2^2) +1] } (process of cross multiplication) and am lost there on how they are actually equal or canceling things out correctly to show xsub1 and xsub2 are equal. but that is what I was told as one way to do.(for injective)

The second way was to graph a few points and show that it is strictly increasing which proves it is an Injective function and from there take the Inverse of the function and test to see if on the range still applies that f(x) = y... still kind of confused on surjection as the examples dont quite show clearly (to me at least) on how to approach it. (sorry im not understanding the material and trying to learn!)

[nona.m.nona]

From another of your threads, it appears that you are not familiar with function notation ("f(x)"), the difference between equations and expressions, the method for finding inverses, or how to work with radicals. This is material which would have been covered in high-school algebra.

You appear now to be taking a college "proofs" course. However, your degree of confusion (you do not recognize that the "Note" provided earlier is exactly the method you have been taught, and the method which is illustrated in the links provided) indicates that you have not studied the content needed to be prepared for this course. In other words, you appear to have major gaps in your background and to need intensive re-teaching.

Please consider having a serious conversation with your academic advisor, as you appear to have been improperly placed in a course beyond that for which you have currently been prepared. Unfortunately, it is not reasonably feasible to attempt here to teach the various courses of material which you need in order to understand your current material.

Either you need to back up and take courses covering the missing information, or else you need to hire a tutor who can work with you, face-to-face, to provide you with the weeks or months of missing content. And your advisor should be able to help you in this regard -- perhaps even advocating for a refund of fees paid for the inappropriate course enrollment.

My best wishes to you in your studies.