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January 29th, 2017, 09:53 AM  #1 
Member Joined: Aug 2011 Posts: 42 Thanks: 0  Is a function's inverse always injective?
Hey! Assume a function $\displaystyle f$ is defined, and we know that $\displaystyle f^{1}(a)=x$ and $\displaystyle f^{1}(b)=x$. Can we then say that $\displaystyle a=b$ because $\displaystyle f^{1}$ has to be injective. I'm thinking that if $\displaystyle f^{1}$ is not injective, then we would have problems defining $\displaystyle f$. 
January 29th, 2017, 10:03 AM  #2 
Senior Member Joined: Sep 2015 From: USA Posts: 2,174 Thanks: 1143 
if $f^{1}$ isn't injective then $f$ isn't a function

January 29th, 2017, 10:25 AM  #3 
Member Joined: Aug 2011 Posts: 42 Thanks: 0 
Right! That's what I suspected. Thanks for the clarification! 
February 4th, 2017, 05:30 PM  #4 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,261 Thanks: 894 
Note that not every function is invertible. The function itself has to be injective in order to have an inverse. And, of course, the function is itself the inverse of its inverse.


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function, functions, injective, inverse 
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