show the map is a bijection

Let , so:

Show this is a homeomorphism:

To show injectivity: We want to show that:

So if we set:

Then from the third equation we get , and can sub back in to find that .

The surjectivity is where I'm getting stuck. If we let any be a point in

Then we get and subbing in to the equation for y means that we get .

So I guess I go on to say that given any I can find a such that

However, I think this is wrong since what happens with the value in this case? I mean we also get that

I'm lost here can someone help? I don't think what I did initially was enough?

Thanks!

Re: show the map is a bijection

I knew something was fishy with this. Check and see that the point (2,2,2), for example, doesn't belong to the image of the function.