Say you have \(f(x) \) and \(g(x)\) and \(g(x) = f^{-1}(x) \).

I observed that these two curves need not intersect, for example with \(f(x) = e^x\) and \(g(x) = \ln x \) never intersecting each other.

I also observed that a function can either have one, two, or three intersections with its inverse, but I was unable to find a function which has more than 3 intersection points with its inverse.

How would I prove or disprove the hypothesis that an elementary function and its intersection can only have up to 3 intersection points? Any counterexamples are appreciated!

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## Comments

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TopNewestConsider \(f(x) = x\).

Note that \(g(x) = f^{-1}(x) = x = f(x)\).

So, there are infinitely many intersections!

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Good solution. I am aware of the infinite intersections solution, but does anyone have any functions which have 4 or more intersections with their inverse (but not an infinite number of intersections)?

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Consider, over any finite interval say \(X\), \(f(x) = x + \sin x\).

Over the interval \(X\) there are finitely many intersections. The exact number depends on \(X\) itself. But you can have

anyfinite number of intersections.Log in to reply

At first, I was unsure of how to find the inverse of that function, so I decided to graph it to verify your claim and you're right!

http://www.wolframalpha.com/input/?i=find+inverse+of+f(x)+%3D+x%2Bsin(x)

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Nice solution

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