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!
Consider $f(x) = x$.

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

So, there are infinitely many intersections!


  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)?


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.

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

http://www.wolframalpha.com/input/?i=find+inverse+of+f(x)+%3D+x%2Bsin(x)
A function can have over $3$ intersection points with its inverse relation.

For example, take $f(x)=x^4-3x^2+2x-3$.

Finding points of intersection of $f(x)$ and $f^{-1}(x)$:



So, there are $4$ points of intersection. $4&gt;3$, thus disproven.