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January 29th, 2017, 08:53 AM   #1
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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 $\displaystyle f$.
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January 29th, 2017, 09:03 AM   #2
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if $f^{-1}$ isn't injective then $f$ isn't a function
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January 29th, 2017, 09:25 AM   #3
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Right! That's what I suspected. Thanks for the clarification!
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February 4th, 2017, 04:30 PM   #4
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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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