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December 21st, 2017, 12:27 PM  #1 
Newbie Joined: Nov 2013 Posts: 6 Thanks: 0  Noninvertible* Functions
I've noticed that the inverse of f(x)=x^2 is NOT itself a function, but can still be described as f(1)(x)=+/sqrt(x). More or less, f(1)(x)=sqrt(x)&sqrt(x). Is there a name for situations like this, where the inverse of a function isn't a function itself but is a concatenation of functions? What about cases like the inverse of cosine, where the answer is more of an infinite family of answers than a single answer?
Last edited by Gigabitten; December 21st, 2017 at 12:30 PM. 
December 21st, 2017, 12:39 PM  #2 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,663 Thanks: 2642 Math Focus: Mainly analysis and algebra 
Context dictates which solutions we take.

December 21st, 2017, 02:52 PM  #3 
Senior Member Joined: Aug 2012 Posts: 2,321 Thanks: 714 
In complex analysis there's a thing called a Riemann surface in which all the possible inverse values are taken as a single geometric structure. The Wiki article has some nice pictures of the Riemann surface for $\sqrt{z}$ and other familiar functions in which you can see what's going on. Last edited by Maschke; December 21st, 2017 at 03:16 PM. 

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