A Function domain/range question

[L Huyghe]

Homework Statement

Give an example of a function whose domain equals the interval (0,1) but whose range is equal to [0, 1].
2. The attempt at a solution
I can't see a way how such function would exits. I though about it this way, if it was the opposite Domain [0,1] and range (0, 1), we could make values 0 and 1 into any number in between without it not being a function, and still having a range that is an interval. If I take domain (0, 1) and produce range [0, 1], I would need to take one of the numbers from (0, 1) and make it [0, 1]. Meaning its no longer has an range that is an interval. It could have a range that is an interval if one of numbers (0, 1) had two possible answers, but the example would no longer be a function.

 

[hgfalling]

You don't have to have an equal-measure correspondence between your domain and range.

Try mapping the domain (0,1/2) to the range (0,1). You can do it, right? Now extend the domain to (0,1) and use the endpoints too.

 

[HallsofIvy]

There is no continuous function that will do map (0, 1) to [0, 1]. I frankly don't see any way to construct a function to fit these conditions that would make sense in a precalculus class. I would separate (0, 1) into rational and irrational numbers, map the irrational numbers to themselves, the "shift" the rationals to fit 0 and 1 in.

 

[Hurkyl]

There is no continuous, invertible function that maps (0,1) to [0,1]. However, there is no continuous function that maps [0,1] to (0,1).

The problem is, IMO, rather easy as soon as you get past the mental blocks -- e.g. limiting your thought to invertible functions, avoiding piecewise-defined functions, trying to do the whole problem in one shot rather than breaking it into easier pieces, et cetera.

 

[vela]

Consider the function y=x². It maps the open interval (-1,1) to the half-closed interval [0,1). If you understand how the half-closure comes about in this case, you should be able to figure out a suitable function for your problem.