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Inverse Functions and their intersection points

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!

Note by Oli Hohman
6 months ago

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Consider $$f(x) = x$$.

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

So, there are infinitely many intersections! · 6 months ago

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)? · 6 months ago

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 any finite number of intersections. · 6 months ago

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) · 1 month, 3 weeks ago