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Why is it that to find the inverse of a function we only need to find if it is one to one.From Wikipedia,

A relation can be determined to have an inverse if it is a one-to-one function.

If we come up with an example in which the function is one to one but the range is not completely utilized(i.e no input produces that output or say it is not onto) then what do we do with that element in its inverse function?

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The issue with being 1-1 is that if you have $f(a)=f(b)=c$, then the inverse function at c would have as images both a and b –  gary Sep 11 '11 at 17:02
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This goes back to the question of whether a function needs to be everywhere defined. If being everywhere defined is a requirement, the function must be one to one and onto in order to have an inverse. You can restrict the set you consider to be the range so that the function is onto.

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