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Exercise [16.08] b http://www.roadtoreality.info/viewtopic.php?f=19&t=1651 
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Author:  Roberto [ 23 Aug 2010, 13:38 ]  
Post subject:  Exercise [16.08] b  
I attach an alternative solution, in which equation N=1/2((a+b)^2+3a+b) is explicitly solved finding a(N) and b(N).

Author:  robin [ 25 Aug 2010, 06:02 ] 
Post subject:  Re: Exercise [16.08] b 
That's quite an interesting way to tackle it. I didn't even try to explicitly find the inverse. But don't you also need to show that the inverse mapping is surjective? Otherwise the inverse you found could be mapping to a subset of NxN and then f would not necessarily have to be 11. For example f: R > R+ given by f(x) = x^2 is not 11 even though it has an inverse g: R+ > R given by g(y) = sqrt(y) (which is not surjective). 
Author:  Roberto [ 30 Aug 2010, 10:27 ]  
Post subject:  Re: Exercise [16.08] b  
The idea was to demonstrate bijectivity of f by finding its inverse g and showing that the compositions fg and gf are the identity; this ensures that both f and g are surjective and injective. In your example gf is not an identity: gf=sqrt(x^2)=abs(x)!=x , and this reveals that g is not surjective (and f not injective). But I understand I was not so much explicit about that, and the free use of auxiliary functions made that even less obvious; so I attach a more explicit version of my solution.

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