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Let's have an invertible function $f$. Let's consider the intersection of this function and its inverse. The following hold:

1) If $f$ is strictly increasing, then the intersection points lie on the $y=x$, which means that one could write: $f(x)=f^{-1}(x)\Leftrightarrow f(x)=x$

2) If $f$ is strictly decreasing and odd, then the intersection points lie on the $y=-x$, which means that one could write: $f(x)=f^{-1}(x)\Leftrightarrow f(x)=-x$

3) If $g(x)=x+f(x)$ is one-to-one, then the intersection points lie on the $y=x$, which means that one could write: $f(x)=f^{-1}(x)\Leftrightarrow f(x)=x$

Is there any online resource (not books, but articles, free e-books, etc.) where each of those is mentioned? I haven't been able to find any resource that contains all of the above mathematical results.

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  • $\begingroup$ By intersection of function and inverse, do you mean points where $f(x)=f^{-1}(x)$? $\endgroup$ – астон вілла олоф мэллбэрг Mar 27 '16 at 10:58
  • $\begingroup$ @астонвіллаолофмэллбэрг Yes, exactly that (if there are any). Actually, those 3 conclusions lead the equation to a more simplified equation. $\endgroup$ – Jason Mar 27 '16 at 11:02

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