Inverses, one2one, onto functions

What's an example of a function f(x) such that g(f(x))=x for some g but there is no h such that f(h(x))=x?
I came up with a proof that showed that there is no such function f, but I relied on the fact that a function that is one to one has an inverse. Apparently a function must also be onto. What is the definition of inverse and what guarantees the existence of an inverse such that f(g(x))=g(f(x))=x?
What function is one to one but not onto and does not have an inverse?
 
A function is nothing more than a line, but instead of Y you use F of X, it's all pretty stupid if you ask me
 

Hurkyl

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What function is one to one but not onto and does not have an inverse?
ex as a function from R to R is a simple one. If f(x) is the inverse to ex, then what is f(-1)?

This also demonstrates a property of 1-1 functions; if you restrict the range, you can make an inverse. In this case, if we view ex as a function from R to R> (the positive reals) then ex does have an inverse; ln x.
 
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Inverses

Do all functions have inverses? (If you place restrictions)
Do all graphs have inverses?
 

Hurkyl

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Do all functions have inverses? (If you place restrictions)
Trivially yes... I could restrict the domain and range to a single point! (though, usually, you can get useful results without such a harsh restriction) For differentiable functions, you might want to look up the inverse function theorem.


Do all graphs have inverses?
TMK, The term "inverse" doesn't apply to graphs.
 
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Re: Inverses

I asked three other teachers at my high school: Do all functions have inverses? Responses: I don't know, maybe, and no because of problems with complex numbers. I looked up the theorem of inverse functions with regard to derivatives and am satisified with that. So your answer to the questions is yes, if you aren't concerned with restrictions? Thanks
 

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